Early on Disc 2, you may choose to grab the Key from Valna and Vigna (the two little girl mummies) without having their dolls (accessible later in the game). None of your attacks will have any effect and you'll be forced to escape unless you want to die.
At the very end of Disc 2, while you're in the Church Nave area, a Gargoyle appears and manages to separate Koudelka from Edward and James. A fight begins, but none of Koudelka's attacks reach the target. The only way is to escape and, optionally, take it on later when reunited with your friends.
In Chapter 3 of the second game, you can't beat the Iron Clefts the first time you fight them, not even with techniques that normally work on their species. After they beat you, you get a new party member, a Yoshi, who comes with an ability that can damage them.
Finally, in Chapter 4 of the second game, you have multiple encounters with a ghost who has stolen your identity (don't ask). Until you discover his name, you can't hurt him, fortunately, unlike most examples in the series, he can't hurt you either. His name is the same every time, but interestingly enough the same place you find his name is the same place you find the missing letter in the name-entering screen. This is done on purpose, because he asks you to guess his name earlier in the game, but people who have already played (or read a guide) know his name when Mario shouldn't. So they removed a letter from the name-entering screen so whether you know his name or not, you can't enter it until you find out his name in-game. Trying to cheat by replacing the missing letter with its uppercase counterpart is the same as not entering the right name, too - you have to go on with the game like normal.
The Shadow Queen is basically invincible until Peach intervenes,
In Mario & Luigi: Partners in Time, Mario and Luigi must fight three Shroobs near the beginning. However, the Shroobs are aided by a giant bomb-carrying UFO that destroys the brothers after three turns, making the fight impossible to win. The intention is for Baby Mario and Baby Luigi to then come to the rescue of their adult counterparts, the UFO being removed in the second fight.
In Mario & Luigi: Bowser's Inside Story Bowser's first fight is with Midbus in which you learn how to fight. Then Bowser starts feeling weak because of the Vacuum Mushroom and the battle ends prematurely.
Saturos and Menardi at the beginning of Golden Sun have attacks that easily hit for twice your max HP in damage. (As a side note, if you win the battle somehow, the cutscene afterward will ignore your victory and still show Isaac and Garet on the ground in defeat.)
The first boss of Super Mario RPG is Bowser, establishing a trend for later RPG entries in the series. Bowser can't be defeated through force, and will eventually whittle down Mario's health through Scratch Damage. Instead, Mario is told to aim for the chain of the chandelier Bowser is standing on, which will end the battle when the Kinklink takes too much damage and drops him.
Neclord appears at Warriors' Village and fights the team. He is immune to all attacks and spells, so all you can do is wait for him to kill your team. After your team undertakes a sidequest to a temple to acquire the Star Dragon Sword, you can fight him at his castle and actually win after striking him with Viktor, who wields the SDS for the rest of the game.
The mass battle against Milich Oppenheimer - whatever move you choose to make, he will release poisonous spores and almost wipe out your army of Redshirts in one go, forcing you to break off the attack and go on a quest to find an antidote against the attack.
Suikoden II: You fight Neclord three times, the first two of which are Hopeless Boss Fights. The first time is in the ruins of North Window, before you have the Star Dragon Sword (which you immediately go to get afterwards, but Neclord flees when you confront him in North Window a second time). The second time is in the mines of Tinto, where the hero and his sister Nanami encounter him. In both fights, Neclord is again immune to everything, but instead of having to die, the goal in this game is merely to survive for a few rounds, after which the battle will automatically end. The third fight is a normal boss battle, where he can be killed.
Suikoden IV: Your first fight is against Glen and Katerina, two very hard bosses. They have over 6000 HP each and you only have 4 turns. You either survive the turns or lose, you'll pass. They can't be taken down with a gameshark.
Suikoden V: In an early boss fight that isn't actually that hard if you use some healing items... but, of course, even if you win, the game acts like you lost. This battle is made harder by the fact that you just came off of another boss battle: you may have already spent significant resources against that one.
Suikoden Tierkreis: There are several boss battles where you only have three or four turns to chip away at near-invincible enemies before the plot kicks in again...
Inindo: Way of the Ninja throws this at you as the first battle of the game.
Naturally, the Final Fantasy series has an abundance of examples:
At the very start of Final Fantasy II, you get beaten senseless by a band of Black Knights. Different versions vary, but whether you see a cutscene or not, you're on a path to a battle you can't avoid, can't run from, and can't win.
In Final Fantasy III, you have to fight Bahamut, normally an end-game boss, very early in the game. During this fight, he has 65536 hit points and heals completely at the end of every turn. The player is expected (and encouraged) to run. If you somehow manage to defeat him, though (by using a cheating device, usually) you are actually penalized; you get nothing for that fight, and he no longer appears later in the game, making it impossible to get him as a summon.
Also from the same game, the first battle against the Cloud of Darkness.
Also, early in the game, if the player tries to move too far into the sea without completing a certain task, they will encounter the Dragon Nepto, whose sole purpose is to block the player's path. However, the game ends normally after the party dies and it's impossible to run from him either, forcing the player to reload the game. It is, however, possible to beat him with massive overleveling, but once you do so, you will be moved a few inches back and the boss will still be there, blocking your path.
Final Fantasy IV also featured a weird version of the trope; the Dark Elf seems to be a Hopeless Boss Fight, but once he defeats you, you get the "Game Over" screen. You have to get a special item from Edward in order to fight him normally. If you have said item, the first fight with him is a Hopeless Boss Fight, but then you fight him again as a normal boss.
With sufficient Level Grinding (or by cheating) it is possible to win the majority of the hopeless boss fights in Final Fantasy IV. If you defeat Kain the first time you fight him, the game pretends you lost. If you defeat the Dark Elf without the Twin Harp, it sets the game back to before the battle. The only fights that are unwinnable under any circumstances are those in which a) the player does not have control of the characters, or b) the fight ends regardless of how you're doing.
The sequel has the Mysterious Girl fill this position several times throughout the game. Also, the fights against Dark Cecil and the Mysterious Girl and Bahamut are Hopeless Boss Fights that result in game overs if the player doesn't have the correct party members. Of course, the game doesn't tell you this.
Final Fantasy V has an inversion in the first battle with Exdeath; it's hopeless for the boss, not the player. Even though Galuf's HP can go down to zero, he can't be defeated until Exdeath falls. Unfortunately, straining his body like this causes Galuf to die immediately following the fight.
At one point in Final Fantasy VI, the game requires one character to stay in a certain town until they fight a certain character. In this case, the Hopeless Boss Fight Guardian attacks the party if he tries to leave, activating what it calls "Guard Mode". No attacks can damage it, and it counters with a powerful blow — the only option is to run. In the final dungeon, the Guardian attacks again, but this time it activates "Attack Mode" and actively battles the party. By now, most players will gleefully accept the opportunity to destroy it. The Guardian also appears in Vector if you go there when you aren't supposed to. Basically, the Guardian is the Border Patrol subtrope played straight.
There's also another Border Patrol in the same town- the Heavy Armor. You actually can beat this one, as it's just a standard mook from slightly later in the game when you have more characters, but only if you are over-leveled. There is no real reward for doing this, though - the only purpose this enemy placement has is to force you to solve a puzzle segment.
Another example occurs in the second half of the game, when you find Terra in the ruined town of Mobliz. While you're suppose to lose the first time the boss attacks, it's possible to kill the boss by exploiting a glitch with the Useless Useful Spell. The end result doesn't change either way, though.
You can exploit another, lesser known glitch (Echo Screen+Joker Doom) on the Guardian as well.
Final Fantasy VIII features a robot that chases the party during an escape sequence. Every time it catches up with the party, it has to be damaged past a certain threshold causing it to enter "repair mode", which allows the player to flee. It can be defeated for real, but only by abusing its elemental weakness and grinding certain spells. If you do manage to destroy that spider, you get an extremely high score on your SEED combat ranking and a whopping 50 AP bonus (the game's second type of experience), rather than nothing at all happening.
And it's relatively simple to beat if you played cards beforehand to gain some good magic, and then mashed away with Squall and Zell's limit breaks. The only time it's impossible to beat is during the first encounter.
Likewise, the Deling City assassination mission in the same game has a boss that's not hugely tough, but the only difference between winning and losing is that winning gets you 20AP
Final Fantasy IX plays with the trope with the Beatrix battles. It is entirely possible to get a Game Over if she kills everyone present (and she likes to spam Shock, which is a 1-hit kill barring Level Grinding and awesome gear), but you can't defeat her. Once you take out her allotted health pool for the fight, she fires off Stock Break or Climhazzard (ironically, both cost less mana and do less damage overall than Shock when she or Steiner use them under the player's control) and reduces everyone to 1 HP before leaving. One useful thing about these fights is that she always has some very nice pieces of equipment for Zidane to steal off of her before she ends the battle (which may take a few attempts at the battle depending on your RNG luck, because the Steal skill is notoriously ineffective).
There also when you fight the game's Big Bad, Kuja. At first it looks like you're going to beat him (prior to fighting him you beat someone that he was confident he couldn't beat), but then he activates Trance and nukes everyone with Ultima.
Final Fantasy X has Geosgaeno, who Tidus first encounters shortly after being transported to Spira. Its physical attack only does percentage-based damage and it has no other attacks, making it seem like a foregone victory - until it tries to inhale Tidus, who then wisely decides it's time to get the hell out of there. Later on, you can fly back to Baaj Temple for a rematch, which this time is winnable if you have the right Abilities on your armor, such as Stoneproof, because this time around his physical attacks have the StoneStrike ability which will petrify and shatter your character, removing them from the battle entirely.
Early on in Final Fantasy XII the party is attacked by Ba'Gamnan and his crew. If you've been leveling normally, it is extremely difficult to win this fight, and the game makes it clear that you're supposed to run away from them (all the way back out of the dungeon you've been going through.) You fight them again later as a mark. Interestingly, if you've done enough level grinding (or gotten everyone a Quickening), you can beat them here, but you still have to leave the dungeon manually - and the game still shows your guest party member running on ahead of you at each zone transition as though you were running away. On the plus side, you get a short scene of Ba'Gamnan breaking his weapon in frustration before running off if you do manage to beat him.
Final Fantasy XIII has a rare case of this and Heads I Win, Tails You Lose in the same fight. Snow fights a bunch of soldiers alone, and although it's quite manageable (and probably easier than the boss battle directly afterwards), you don't get a game over for losing, nor do you get anything of value for winning. Either way, the following cutscene is the same.
Another example comes from Hope taking a level in badass right after Snow practically broke his back saving him. The boss mob you fought not fifteen minutes earlier, with BOTH Hope and Snow fighting fit, comes over and incapacitates Snow. Hope then takes on the boss by himself! There's no way to win this fight alone, though; whether your HP drops to 1 or you fight the boss for an extended amount of time, Lightning and Fang come to the rescue.
Crisis Core ends with one of these: As Zack, the player is forced to fight his way through an unlimited amount of Shinra infantrymen (they will not stop coming, the battle is scripted to end first), who then leave Zack bloody, bruised and barely able to lift his sword at the end. Then three more Soldier infantrymen show up and pump him full of lead, similar to the God of War II example below (except that he doesn't get better).
Dirge of Cerberus has a fight with Weiss the Immaculate, who initially moves so fast he's nearly impossible to hit. He also takes a VERY small amount of damage, but it may be theoretically possible to beat him. After Vincent is defeated, he gains enough control over his Super-Powered Evil Side to be able to keep up. The following battle is on more even footing.
Not so much. Weiss the Immaculate's health bar is static: it doesn't change at all, no matter what damage you deal, even if it is far over his own max HP, which is 21635. If you manage to put up enough of a fight to deal that much to him, his health will never go below 1. If you hack the game to make that happen, he continues on fighting, but the thing is that your bullets clip right through him, as will any other attack. You really can't win, no matter your skill.
Dissidia 012. If, at the beginning, you answer that you're a "Dissidia expert," it will make you fight the superboss version of Feral Chaos with your singular preset level 1 character. Overlaps with Trolling Creator, because you're likely trying to avoid unnecessary tutorials because you've played the original, and this doesn't actually prove that you lack knowledge, just the levels and equipment to make winning probable.
This is a real pain if you don't know it's coming (wasting healing items). This is also true in the Japanese version; you always use a Phoenix Down when you get killed, and this boss is no exception. Thankfully, they fixed that.
World of Final Fantasy has a few throughout the game. The first happens right around the beginning as a means of demonstrating that losing a battle normally will send your playable characters back to Nine Wood Hills. There are a couple more in the No-Gear Level, D-District Prison.
Many times in Radiata Stories, whose system is such that the bosses secretly have infinite HP but otherwise fight on what appears at first to be equal footing with the player (only doing about as much damage per hit as normal enemies, etc.) However, after hitting the player about 5-10 times, they suddenly unleash a Limit Break that is unavoidable and (on normal playthroughs) invariably powerful enough to wipe out your party in a single shot. With a cheating device (or on a New Game+,) the game simply flat-out doesn't care if you're actually able to take the blow and survive—your characters simply fall over and die after being hit with the attack, even if they still have over 3/4 of their health remaining.
Valkyrie Profile gives one either the option of immediately attacking the King of the Vampires, Brahms; or listening to what he has to say. Plot-wise for the Best Ending you should listen to him. Fighting usually results in him kicking your butt and scolding you which results in a decrease in the Karma Meter. However, if you actually beat him, the game just ignores that the battle ever happened (although you do get the karma bonus). And if you get the "C" ending, Freya comes down and kicks your ass.
The same happens in Valkyrie Profile: Covenant of the Plume if you use the plume too much and during two of the endings. For the "A" ending, the initial attack against Garm is unwinnable, and all you have to do when you fight the Valkyrie in the "B" end is reduce her to half (or less) HP.
In Lufia & The Fortress of Doom, Gades is unbeatable. In following games he is meant to be lost to, but is beatable with some effort and over-leveling. Defeating Gades in combat would result in a rare equipment drop, and a slightly altered cut-scene afterward where the Sinistral blinks, then uses non-combat cutscene powers to defeat your characters anyways.
In the first game, it is truly impossible to defeat the first Sinistral, Gades, when you first meet him (not counting the prologue). It's something of a tradition that this guy defeats you when you first him. Amusingly, this guy is depicted as weaker than others in the next games, but in this game he appears to comparable in power (some people say he's actually the hardest).
Lufia: The Legend Returns has lots of these. The first few battles with Gades, the Ancient Dragon M, the first few battles with Daos...of course, as mentioned above, if you do manage to win a Hopeless Boss Fight, you get a sweet item, then the boss kills you in a cutscene.
At one point during the main storyline of Chrono Trigger, you are unexpectedly pitted against Lavos, the final boss. If you've leveled up normally, you will probably lose instantly. He will surely attack first, and will use a Total Party Kill attack that, assuming you leveled up normally, is instant death. Just to make sure you lose, he's actually even stronger than the normal Lavos at the end of the game. It is actually possible to beat him, but you'll need to be in New Game+ and leveled pretty high for there to be any realistic chance of that. Disappointingly, even if you win, you don't get to see Queen Zeal's reaction to the "invincible" Lavos being defeated right in front of her. It leads to a secret ending, one of the most difficult in the game to earn.
In the Kingdom of Zeal, the plot requires you to be captured. To this end, a Golem is sent to attack the party. While the standard fare would be to lose to the monster, you can actually defeat it (in fact it can be pathetically easy if you know how). Doing so results in the summoner using the "look behind you" trick then shooting you in the back. All winning does is net some decent EXP. What's worse is that, later on, the same guy will summon two of these things that you are expected to defeat, so unless you figured out how the first time around, good luck.
If you go to the Northern Ruins in 1000 AD, you'll encounter Cyrus' ghost. Physical attacks always miss on him, and he's immune to all elements so magic doesn't work either. The battle ends automatically after a few rounds with the party realizing they can't beat him. The correct thing to do here is to go to the same area in 600 AD and Set Right What Once Went Wrong.
In Chrono Cross, Lynx switches bodies with Serge in the middle of the first fight against him. Serge (as Lynx) must then fight Lynx (in Serge's body), Kid, and one other party member. The battle is meant to be impossible to win, but if you manage to beat them (probably by virtue of being in New Game+), there's an additional cutscene. Serge walks over to his own body, then Kid jumps up and stabs him before collapsing again. The story will then continue as if you had lost.
The tutorial battle against Radius at the beginning of the game will end with Radius wiping Serge out with a single attack, which he spams if you somehow survive. Even in New Game+, no matter how much damage you do, Radius won't fall. However, if you manage to last enough turns, Radius compliments you and simply gives up.
Similarly, the two Big Bads in Skies of Arcadia, Galcian and Ramirez, have infinite HP if fought at any point besides their final appearances. The correct options are, respectively, to outrun Galcian and surrender to Ramirez when given the choice.
Also, the Red Gigas is completely invincible when you fight him, forcing you to just survive until a scripted event. The Green Gigas does have a limited life bar, but such a ridiculously high HP count that the game expects you to immobilize him instead. However, neither fight actually requires that you lose, making these a borderline case note The Red Gigas takes damage but it's just that he has such ludicrously high HP that after multiple Harpoon Cannons and combination attacks using torpedoes, subcannons, AND main cannons, he's taken just enough damage to his health that the difference is actually visible. Similarly, there is an odd glitch when facing the Green Gigas the first time, which prevented the immobilization from taking place, and as such had to knock his HP to zero to kill him. The immobilization cutscene occurred anyway.[[note]].
Ramirez can be fought if you either cheat or overlevel. If you're strong enough to withstand his ultimate attack (which after a while he starts spamming over and over again), and are willing to spend several hours fight him, you can get his HP down to 25%.... whereby the game glitches up and goes to the cutscene where you lost. Talk about hopeless. Ironically, if he had his final boss stats (where he's supposedly stronger, storywise), he probably wouldn't even be that hard to beat.
Ramirez has 9.999.999 HP in those unwinnable battles. Worse, in the first battle against him in Nasrad he can cast Sacrulen (a FULL HEAL on himself) if you hack the game to reduce his HP.
In Tales of Destiny, Leon is nearly impossible to beat the first time you face him, which is quite early in the game. If you do manage to win, you get a special ending right then without playing through the rest of the game.
In the PS2 remake, however, your efforts are rewarded by him suddenly unleashing his Limit Break and ending the battle regardless of whether everyone actually got KO'd or not.
The first fight against Shizel in Tales of Eternia. She'll give you the illusion that you're actually fighting her, and after a little while, she opens up a can of god-power that knocks everyone down to 1 HP before effectively flicking everyone to death. (Bonus points if, instead, the A.I. Roulette decides to finish you off with something powerful like Prism Flasher.) If you manage to pull yourself back from the brink of defeat, the game cuts away from the battle abruptly and pretends you lost anyway when she's reduced to half HP.
You can actually avoid losing the first fight against Yggdrasil (though that isn't likely because if you lose the scripted fight prior to that you will start the fight off with 1 HP), but it is virtually impossible to beat him because the fight will automatically end after a certain amount of time. However, using certain New Game+ upgrades, you can win the fight and force him to flee. He then pulls out Cutscene Power to the Max, giving you a glimpse of his nifty Sword of Plot Advancement in the process, which justifies both the trope and the rescue cutscene immediately afterward.
The first battle with Vidarr has a feeling of this trope included as well. Although you still have to win, no matter how well you do, your team will get a cutscene with them about to be killed, only to be saved by Kratos. You can Action Replay this, hit him once down to near death, but this scene will trigger regardless.
The party's first battle against Vaclav in Tales of Legendia. He only has about 8000 HP, but once you've worked him down to about 4000 he unleashes a massive spell that knocks your entire party unconscious.
Tales of Legendia flat out spams this trope. You get stuck with unwinnable fights in chapter 1, chapter 3, chapter 5, and twice in the last chapter of the black mist story arc.
Funny thing about the unwinnable fights in Legendia, is that the fights you are supposed to lose, is that the attacks the boss "defeats" you with actually do 0 damage.
Tales of Hearts features two boss fights you're "supposed" to lose, the one against Chalcedny early on for Kohak's Shy Spirune and the one against Sorceress Incarose about halfway into the game. Both of them are technically winnable, and in a subversion, if you beat Chalcedny, he actually does hand over the Spirune immediately (though the party finds another excuse to go through the area where you'd normally actually get it back).
Symphonia's sequel, Dawn of the New World, has so many of these, that the player may have trouble discerning whether a given boss battle is winnable or not (on his or her first playthrough anyway).
The game also has two types of these: Fights that are just really, really hard to win, and fights that you can't win at all (because the boss' HP won't go down). This can become downright ridiculous, like the first fight against Commander Brute. Cue a long string of "Impressive !" as he flinches constantly under your attacks... without losing any HP. During the second encounter, he doesn't get any stat buff but loses that plot invincibility. Curb-Stomp Battleensues.
Lloyd, the first boss of the game is normally unbeatable, but if you can carry your strongest monsters over to your second playthrough or know how to trap him in an infinite combo, he's beatable, thus netting you several levels of experience. But the story advances as if you lost.
Tales of the Abyss has a fight where Luke fights Asch alone that is expected to have you lose. You can win, but it requires you to be either very skillful with the battle system, level grind, be in New Game+, or have a lot of healing items. There is a slight variation in the cutscene after if you do manage to win, and you do get experience, so it's not all for nothing at least.
It's pretty entertaining to hear Asch whine and cry about how worthless he is because he just lost to his replica.
Not really related, unlike other scripted fights in the series, the scripted boss enemy is actually weaker the bad guys he's supposed to equal in rank, but it evens out since you fight him with only one character.
The original version of Tales of Vesperia didn't have one of these, but the PS3 version adds a one-on-one fight between Yuri and Don Whitehorse at the end of Keiv Moc. It is possible to defeat him, even on the first playthrough...but only if you've undergone some serious Level Grinding, set the difficulty to Easy, and are very, very dedicated - otherwise, you'd be best advised to try again the second time around. Nothing changes much if you do win except for acquiring an okay-but-not-great accessory (a Diamond).
In Tales of Graces, the first fight against Hubert in the adult arc is this. Even if you're either leveled enough to deal decent damage against him, or skilled enough with the game's combat system to dodge his very wide close-range attacks, he'll build up his eleth gauge incredibly quickly and then use his Mystic Arte to end the battle. Even if you get strong enough to deplete all of his HP, it'll simply display as ???? and he'll keep on fighting.
Tales of Xillia: When the party fight the real Maxwell it consists of 4 rounds with cutscenes in between, the first three you can't beat Maxwell no matter how hard you try, as he'll eventually fire off a party wiping attack each round (which ironically if the party is equipped well hardly does any damage) which leads to the next cutscene which they regain their feet and charge again. It isn't until Jude's Did You Just Punch Out Chtulhu moment and Milia joins the battle can you win.
Tales of Xillia 2 opens the game with one of these, and it'll only end if you get knocked down (Which leads to the boss hitting you with his mystic arte). Due to an oversight, the fight can potentially never end in a New Game+, as if you have the Glory skill equipped, you'll never get knocked down.
The second phase of the second battle with Khronos is also meant to be such, as you aren't given any clues as to how to prevent him from using his time reversal technique [[note]]Just activate Ludger's Corpse Shell when he tries to use it, which will result in the fight ending if he pulls it off successfully. However, if you manage to stop him, you're allowed to defeat him and get some EXP for your efforts, though the following cutscene is unchanged.
Tales of Berseria: The first fight against Artorius is one of these, both gameplay-wise and storywise. At a time when your party members might be around level 20, he's level 60, and your attacks only do Scratch Damage, if that. Eventually, Artorius unleashes his Mystic Arte on Velvet, causing a long cutscene to trigger where Velvet can't even hit him.
Unusual examples from Baldur's Gate: At the very beginning of the first game, as you flee the training area of Candlekeep, you and NPC Gorion are accosted by an armored figure who will eventually be Sarevok, the final boss of the game. A cutscene follows in which the PC flees and Gorion engages the enemy and gets himself killed. However, the battle is still conducted according to the regular rules of the game engine, and occasionally Sarevok's final swing actually misses — but because the game is scripted that way, Gorion drops dead regardless. If you manage to make Gorion or Elminster hostile to you, you won't live long; their battle scripts makes them cast one Lightning Bolt spell for show and then just executes the kill() instruction on your main character, which is Exactly What It Says on the Tin: an unavoidable One-Hit Kill.
In Baldur's Gate II, you're not even given the option to fight; many boss characters will simply automatic kill you if you advance the plot wrongly. They all wear a secret belt that prevents their hp from falling lower than 1. You can see this belt right at the start of the game, when Imoen is wearing it. As Imoen is essential to the plot later on, she cannot be killed; if she were to fall below one hit point, a script causes her to panic and run away. The exploit that allowed you to kill essential characters was that if you somehow managed to reduce your constitution or intelligence stats to 0, you would die regardless of the hitpoints you had. The ability to reduce constitution only shows up in a few traps and is inaccessible to players, while the ability to reduce intelligence shows up in the game for only a short time; The mindflayers you encounter in the underdark automatically do this on a successful strike (hint: they're eating your brains). There was one item near the endgame that allowed you to summon/transform into a mindflayer for a short time, thus allowing you to reduce intelligence. This exploit is fixed in the expansion, though you probably wouldn't have access to it early in the game without cheating anyways.
In Mega Man Battle Network 3, Bass is almost unbeatable the first time you fight him. He has a shield up that prevents the main character from ever doing any damage. It can even block a chip that's sole purpose is to blow away those kind of shields. Naturally, this shield has become his usual 100-HP aura in time for the real fight (with no plot explanation given).
It was found that it's possible to defeat Bass using the Balance chip and hitting him during his one attack where he drops his aura. The game actually gives a different cutscene for this.
There's a bit of an explanation for the shield turning to usual, actually. Bass had just absorbed part of Alpha's program, which actually weakened him and probably caused his aura to become not-invincible.
Similarly, you can't do anything to ShadeMan in MMBN4 until the second go-round — and even there, you have to let him beat you up a bit.
MMBN1 presented an unbeatable ElecMan, who recovers completely from all major damage and also won't be defeated even if a player is quick enough to drop him to 0 HP. Rather than dying, however, you only need to watch ElecMan heal himself a few times before the heroes learn they need to disable his power and begin the real fight.
While we are talking about Mega Man, we should mention Mega Man Star Force which has four bosses that are unbeatable.
After you beat HarpNote in the first game Geo is tricked by Luna Platz to coming back to school by having him look at the play set (which is about the battle against Taurus Fire) a jammer drops a light on you but misses, after which you have to chase after it. When you finally catch up with it, the jammer ambushes you, you battle and then after 4 turns the battle ends and he strangles you causing your character (Mega Man Geo-Omega) to pass out and un transform.
Then in the second game you go to the shopping center area (where you battled Dark Phantom and later on battled a eyeball monster), while here you are forced by Hollow to save people from multi colored knight monsters. After you beat them you are forced into a 4 round battle against Solo-Rogue after which Sonia shows up and saves you.
Next in the third game, you first have to go up against J Corvus, after 4 rounds the battle ends and Joker shows up and datafies Luna. Then later on you have to go up Acid Ace who has been corrupted by Joker (who else?) and after a 4 round battle (what else?) the battle ends and he un transforms and collapses on the floor of the concert stage at the TV studio.
The battle with Queen Beryl in Sailor Moon: Another Story. Her regular form is perfectly beatable, but upon defeat she transforms into Super Beryl, forcing you to fight her again, twice. During the first battle in that form she has a move that does 9999 damage to each character in your party regardless of their stats, and she almost always uses it on the first turn, sparing you from futile attempts to defeat her.
Breath of Fire III has two bosses, Balio and Sunder, who are fought twice before you can actually beat them. It is possible to temporarily hold them off in the second fight, but the first is completely unwinnable. It was years, though, until this was completely proven. A rumor, fueled by a strategy guide misprint, insisted that there was a prize for winning the fight even though this is impossible. It was eventually found that they don't even have HP values in the first fight - they're just targets you can hit. It's actually possible to lock them in an infinite loop of trying to cast a spell without enough MP and attacking each other.
There's also Garr in the Inevitable Tournament. If you stall too long, he says "Let's finish this!" and defeats Ryu with a super critical.
There's also a few in Breath of Fire II. The very first battle of the game, the second battle of the game (against a giant, vicious demon that later returns as the second-to-last boss, no less) and the optional fight against Tiga late in the game. The first two can be excused, since you're in control of a small child (Ryu can't be much older than 8 at that point, as he seems to be a teenager, 10 years after the fights), but how Tiga manages to keep going is a mystery, especially since he's later CurbStomped by a boss that isn't even that strong.
This is reversed in Breath of Fire IV You instead play the Boss and he is so powerful you can't lose even if you try. You play him so he may not be considered a boss but he still fits, in every other way, the typical role of the last boss. If you beat him by using a cheat device, the game freezes. Right before you fight the final boss, you have a scripted fight with him where he will eventually defeat with an instant death move.
In Grandia, your first battle with Gadwin is a Hopeless Boss Fight, as he takes no damage from any attack, and will eventually attack you with Dragon Cut, which will deal 9999 damage.
The first time you meet Millenia in Grandia II is a Hopeless Boss Fight, as she has infinite health, and the fight only ends when she blasts your group with a lightning spell. Notoriously, this fight can glitch and go on indefinitely with her never casting the spell that ends the fight. This is repeated for a few other battles; it is particularly noticeable in this game because Grandia 2 has a mechanic that normally would allow you to prevent enemies from acting by interrupting their attacks. Naturally, this wouldn't do in an unwinnable boss fight, so instead all such attacks simply and inexplicably fail, the only point in the entire game at which they do so. When you fight the same bosses later—sometimes only shortly afterwards, with no explanation for the change in power or winnability—your attacks start working as normal again.
This sequence of events differs very slightly in the Dreamcast version of the game; Using a cheating device to increase Ryudo's health, speed, and damage potential means the battle can be won by reducing Millenia's HP to 0 - though this is incredibly time-consuming and if she is able to use the lightning spell mentioned, the fight will end immediately. Your counter move also functions as expected if you can land it at the right time. You don't gain any rewards for winning this way, and the game continues as if you'd lost to the lightning spell anyway.
In a similar case, the first battle with Melfice is Hopeless. He takes virtually no damage from your attacks at normal levels for this point in the game and his attacks wreck your party. The battle ends whenever he uses Wailing Sword Slash, in the same manner as Millenia's Zap. However, this one is easier to cheese: do some serious level grinding, use Millenia's Spellbinding Eye attack to freeze Melfice for a time (Cancels don't work on him either, even in the Dreamcast version), then wail as hard as you can at his individual parts. It'll take a while, but Melfice will eventually go down, even going through his death animation...then the game continues as if you lost anyways, just like with Millenia.
In Jade Cocoon: Story of Tamamayu, when the main character fights the Dream Man and his dragon-like minion in his dream, he cannot be defeated. Even if you win (through cheating or pure luck), only the dialogue changes during the scene after the battle, with the main character still lying on the ground as if defeated. You can win by regular methods as well. He's difficult, yeah, and definitely this trope. But precision tactics can defeat him.
In Jade Cocoon 2, towards the end of the training forest, the main character touches Nico's cocoon and is cursed. The subsequent battle cannot be won without cheating, since the Kalma the main character must fight is too powerful for your two newly hatched Divine Beasts to handle. After your inevitable defeat, the main character is "possessed" by the Kalma. However, you take him on a second time much later in the game, and while he is definitely a challenging opponent, he's far from unbeatable.
In SaGa Frontier, the first fight of Red's game, against Shuzer, requires you to get pummeled into the dirt, so Alkarl can show up to save the day.
In SaGa Frontier 2, Wil Knight's team has to run away from a human transformed into a giant beast; this battle is unwinnable regardless of how much level grinding you do. (You can, however, challenge it again if you so choose.) In an earlier chapter, Wil's team fights against two dragons which will kill you under normal conditions, then one of your allies will get back up and commit a Heroic Sacrifice, killing the two dragons and healing the party. If you don't have that specific ally in your party, you lose the fight for good.
Romancing SaGa has this with the Diamond Fatestone. It is possible to win, but very hard considering your levels at the time you talk to Schiele the final time, especially on New Game+ where even after talking to her once after clearing the event in prior playthroughs, the event will trigger.
The old game Might and Magic II had a monster called the MegaDragon in a location reached by Time Travel. It was undefeatable because it served a plot purpose; however, it could in fact be defeated with high level characters and non-elemental spells like Mass Distortion and Implosion. The monster showed up in weaker forms in later games in the series.
In Might and Magic V, there is another unbeatable boss, the Big Bad. Notably, it's not even a fight... if you enter the Big Bad's room without the Big Good in your party, BB eradicates your entire party with a wave of his hand and then you get a game over. If you do, the game goes straight to the ending cutscene of the two of them fighting it out while your party waits outside.
Might and Magic IX had an interesting take on the Hopeless Boss Fight, that may have actually been kind of cool if the whole game hadn't been released as a shoddy rush job. To defeat the game's Big Bad, a god of chaos named Njam the Meddler (sort of an Expy of Loki) the PCs first had to trick him into chasing them to the Tomb of a Thousand Terrors. They then had to make their way through the place, which had deadly booby traps and the most lethal monsters in the game, until they got to the final room, where they could activate a lever that would imprison Njam forever. Of course, Njam was trying to stop them the whole time, and he'd appear out of nowhere at random times. Trying to fight him was suicide, as you couldn't harm him, but if you just avoided him long enough, he'd disappear and you'd have at least a few minutes before he'd attack again.
There's a Hopeless Boss Fight in Phantasy Star IV; if you insult it by acting greedy instead of heroic, but you can beat it with enough Game Genie-ing, though you lose the spell you should have gotten from it in the process.
A similar situation occurs in Phantasy Star II. If you kill the security robots coming to arrest you by cheating, Script Breaking happens and you can't progress through the plot.
Phantasy Star I has what initially seems to be a Hopeless Boss Fight with the "Saccubus", a monster that appears once you fall asleep in the Governer's Mansion and quickly decimates your party at normal levels. Fortunately, it was All Just a Dream. With a bit of Level Grinding, however, it's entirely possible to take down the Saccubus - earning the reward of one single meseta.
The true Hopeless Boss Fight of Phantasy Star IV is actually against Zio the Black Magician, who cannot be defeated due to his Magic Barrier, although further into the game you collect an item that will disable the barrier.
Also in Phantasy Star IV are the carnivorous trees on Dezolis, which are used to block progress until you have the Eclipse Torch. They are unbeatable in that they will continually respawn forever. When they are approached with the Eclipse Torch in your inventory, though, they are instantly destroyed, without a fight.
Wild ARMs: Alter Code F has strange Hopeless Boss Fights. After a few turns, the battle automatically ends — either just closing, or with your party being instantly wiped out — but you still get EXP from the encounter based on how much damage you managed to deal first, so it's still a good idea to go all out.
Straight after the battle with Master Eddy in MOTHER 3, you wash up on a beach with all your items gone, and all your characters at a mere 1 HP and 0 MP. Healing your party is impossible without eating from the nearby patch of hallucinogenic mushrooms (an important plot point for this chapter), and there's an enemy Zombieshroom blocking your path forward which, in your party's current state, is suicide to encounter and should be impossible to defeat. Oddly, after you eat the mushrooms, it's not even necessary to fight it at all, as it'll have moved far enough to the side to be avoided. Cheat and kill this enemy before using the mushrooms, and it will still be standing there in your way after the battle.
Run into the Ultimate Chimera and it will be an instant Total Party Kill. The game doesn't even bother with a staged battle sequence; it skips straight to the game over screen.
The trope is subverted in the battle against Porky. When you reduce his HP to zero, he gets into a capsule that makes him completely immune to any attack you throw at him, making it look like you are about to be crushed in battle. However, the boss also lacks any attacks of his own and after a few rounds, the party decides nothing can be done against him and he can't do anything to them, so they just stop fighting and move on.
Early in Legaia 2: Duel Saga, there is a boss fight that pits the lone main character against a brother and sister, both boss-strength. The battle ends when the player loses and the bosses execute an ultimate team attack. Alternatively, a prudent player can actually defeat one of the bosses, at which point they get angry and get right to the team attack anyway. The team attack doesn't do damage, it just ends the boss fight.
Two dungeons earlier in the game is the battle against The Man With Golden Eyes. He takes zero damage from attacks, and simply watches you until you've attacked several times, then easily defeats you.
A subversion is the battle against the Blue Knight while trying to escape from the castle early in the game. It's a timed battle, and he's very strong, so that it seems like you'll lose if you take too long, but you can't win if you rush him. The timed battle aspect is false: no matter how long you take to kill him, you'll win.
In Knights of the Old Republic, players can trigger an optional Hopeless Boss Fight by pissing off Calo Nord in the Lower City cantina on Taris; since the plot requires his survival, he's invincible here.
For that matter, the nameless Sith mook running the sentry turrets at the elevator from the Lower City to the Undercity? Likewise invincible! Trust me, never choose the 'fight it out' option in the dialogue. Those sentry turrets just don't stop.
In your first encounter with Darth Bandon while escaping the Endar Spire, the fight is so hopeless that you don't even get to try. Your Red Shirt teammate sacrifices himself so that you can escape.
In your first duel with Malak, you can't kill him. You can only hold him off until Bastila shows up and sacrifices herself to let you escape.
Doubles as a poorly implemented example of this trope: Malak is incredibly weak in this appearance, is easily bested and actually flees at one point before Bastila"sacrifices" herself to buy you time to escape, massively lessening the impact of said sacrifice in the process. Doubly irritating since in his Final Boss form, he's way more difficult.
In Knights of the Old Republic II, Darth Sion literally can't be killed. Every time you reach a damage threshold, he becomes invincible for a moment as his health is restored. With the right build, you can keep him on the ropes forever, but pure combat won't ever work. When you meet him on Korriban, Kreia makes you and your party run like hell after a few hits. The second time, you have to talk him to death.
In the Expansion PackShadows of Undrentide, you are expected to do a couple of quests in the Elven Crypts for the spirit of an elven hero. If you complain even once about this (something fairly innocuous along the lines of "Oh great, more fighting...", when rebutting other characters in much stronger terms just scold you before carrying on), he takes mortal offense and attacks you. At the level you're expected to be at (around three at most), you will die.
In the Tournament of Arms, should Claude make it to the final match (he does not have to, but you get a decent weapon that can be upgraded into Claude's ultimate weapon if you do, so it's well worth the effort), he will face Dias Flac in a Hopeless Boss Fight. Dias is invincible here and will take off a set percentage of Claude's HP with each strike, meaning you'll die in a few attacks no matter how strong you are. It should be noted that, with the help of a Game Shark to put attack items in your inventory (all your items are temporarily taken from you before the tournament begins), you can actually defeat Dias, but storywise, the game will still consider you to have lost the fight.
Dias cannot be beaten. You can 'shark Claude to have 9999 STR and CON in one playthrough. Every monster in the game, including the final boss and all the optional bosses, did zero damage to him, and Claude in turn always hit for 9999 regardless of what weapon he had. Dias... two shotted Claude. Dias either took zero damage or outright blocked every one of Claude's attacks. If you include the above poster's cheating in attack items, only the Magic Rock will damage him, but his HP never decreases. That's what makes him Unwinnable by Design.
You actually CAN defeat Dias, sort of... As it turns out, Dias is invincible, but only until the exact moment Claude is knocked out. If you are strong enough and manage to perform an attack just as you die (perhaps Air Slash), you can force a tie. Of course, since you're dead too, it counts as his win. To summarize, just take the consolation prize.
At the Lacour Front Line, you'll face a monster called Shin that is also immune to all attacks. The goal is simply to keep your team alive for about one full minute. Afterwards, the fight will end automatically. A short time later, you'll face Shin again, and he is again invincible. The only difference is that to proceed with the game, you have to let him kill you this time.
At the end of the first disc, you'll meet a group called the Ten Wise Men, and face one of them, called Berle, in battle. This is like the first encounter with Shin (who will ironically be finally beaten in the same area you fight Berle in), in that Berle is invincible, and the battle will automatically end after one minute.
Sometime in the second disc, you'll face another of the Ten Wise Men, Marsilio, who is, like Berle, invincible. This fight is like the second encounter with Shin, where you must let him kill you entire team to proceed.
Finally, in the optional Cave of Trials, you'll encounter a monster called the Weirdbeast on the fifth level. This monster is a Hopeless Boss Fight at first, unable to be killed by your team. To proceed with the dungeon, you must run from the fight, and explore the rest of the floor until you find a weapon called the Weird Slayer. True to its name, the Weird Slayer will destroy the Weirdbeast in a single blow (as well as any other creature with the word Weird in its name). The first Weird Beast fight is technically not hopeless, but the Weird Beast only every takes 1 damage from an attack and has over 800,000 HP. You can kill it with the "Medusa Shield" and "Bubbly Potion" and such are all legitimate since they were used the way they were meant to be used. Either way, they are still fairly difficult to pull off.
Magna Carta mostly avoids this trope; there are a number of ostensibly unwinnable fights throughout the game versus powerful boss enemies which you end up having to beat later, but unlike most such games these fights are actually winnable - it just requires a great deal of strategy (and sometimes some power levelling) to do so. Frustratingly, there IS a single truly hopeless boss fight towards the end of the game, one which you cannot win under any circumstances, though it is fairly obvious that it is hopeless... other than the fact that the other boss fights in the game were winnable.
The first Kingdom Hearts features a fight with Leon early on in Traverse Town that's extremely difficult compared to any of the previous bosses, due to him possessing quick and powerful attacks that can take you out in one or two hits without excessive level grinding. While the intent is for you to lose, he's perfectly beatable if you abuse a flaw in his AI, which will net you some EXP and a slightly different cutscene where Yuffie accuses him of losing his edge.
Later on in the game, Cloud in the Olympus Coliseum is likewise quite tough to beat, but win or lose, you still go on.
In Kingdom Hearts II, any battles where Roxas fights Nobodies without a keyblade count, similarly to Sora attempting to fight Heartless during the Islands' destruction (again, without a keyblade) in the first game.
The first few times you face Hades, he's completely invincible due to him being more powerful in his personal realm of the Underworld. You're only able to fight him proper when you have Hercules backing you up.
358/2 Days has its share. A battle with a boss heartless who turns out to be Xion under an illusion ends after you chip its HP down far enough, and another battle ends after a certain amount of time has passed.
Seen in Kingdom Hearts: Birth by Sleep in Ventus's story when you fight Vanitas for the first time in the badlands, as soon as he is down to one HP the game ends the battle. Later, Micky teams up with Ventus and the fight is winnable.
Wandering Hamster has this trope in the form of Ghost Broaste, who you cannot kill until you find the worthless weapon, his old Teddy-Bear.
In the Magic Knight Rayearth SNES RPG: the first battle against Lafarga, and the first battle against Emeraude.
If you attempt to attack the Big Bad Rhapthorne in Dragon Quest VIII after he's put up his barrier, nothing will touch him, but his attacks will still hurt a lot. To take down the barrier is rather involved, requiring a side quest and a complex ceremony - in the middle of a battle.
The first encounter with Marquis de Leon (Keeleon in the American version) in Dragon Quest IV also counts, since after defeating Balzack, he's powerful enough to easily One-Hit Kill Meena (Nara) and Maya (Mara) before they can even put a dent in him. The battle will end if both of the sisters are knocked out, regardless of how much health Oojam (Orin) has.
In Dragon Quest IX, there are two fights against higher ranked Celestrians, that are impossible to win because lower ranked Celestrians (like the player) cannot act against higher ranked Celestrians, and the player will be prohibited from attacking. While the player cannot win, if they've leveled up enough, they can survive the assault until the boss gives up and leaves.
Dragon Quest VII has two in very close proximity to each other on the Dharma Temple/Alltrades Abbey island. After the party has their magical spells and skills stolen from them, they find themselves in a destitute town filled with other people who've had the same thing happen to them. Strom, the boss of the town decides to beat you mercilessly with his thugs in order to show you who's in charge. Not long after, the party tries to make their way back to regain their abilities with a Guest-Star Party Member in tow, but he abandons you at the last minute, forcing you to fight against another pair of bosses you cannot hope to win against. Another happens later in Hubble, when Hybris uses the Incantation and transforms into a monster - he's not beatable until shortly thereafter.
The fights with Reaper Beat and the Trance Rhino (a Taboo Noise) may feel hopeless the first time through, but cut off after a certain amount of time regardless of who's winning. But if you die before that, the Game Over is for real.
The battle against Taboo Minamimoto feels hopeless as well, as it typically ends with your defeat. However, if you are defeated too quickly, you receive a Game Over instead of the cutscene that follows.
In a New Game+, both Reaper Beat and Taboo Minamimoto appear as Blue Noise symbols and can be fought in optional, standard battles.
And you'll probably fight them over and over. If not for the satisfaction for kicking their asses, then for the sweet pins they drop that you'll need for some of the more useful stickers.
The first boss fight in Metal Max Returns is an unusual version of this. The enemy is far too powerful for you to beat as early in the game as you fight it... but you lose as usual if you die. Instead, if you manage to survive for two rounds, a computer-controlled character joins the fight and will win it for you on the next round; you still get the reward for winning, but it does have some consequences in the plot.
Xenogears has Id and his invincible Gear show up to beat the party into the ground a couple times.
Followed closely by an impromptu fighting test culminating in a Hopeless Boss Fight against eventual party member Rico. Even if you do use a Gameshark, you still can't win. After beating down Rico (which takes a while - he has about 65,535 HP, give or take) the game simply continues as if you'd lost!
Alpha Weltall is a boss you aren't supposed to beat either, but some crafty and Crazy-Prepared players have beaten him, and do so as a sport. Since it does have a drop if you have the Trader's Card (The "Slayer's Robe") it seems the writers had some idea it was possible.
Subverted in Xenosaga in the first fight with Margulis. The fight is very doable even without grinding as long as you counter his gimmick, and while the game continues if you lose, the cutscene afterwards is ambiguous enough that it makes sense whether you beat him or not. Either he has you on the ropes, or you fought him to a standstill; either way, Ziggy creates a distraction with explosives he placed earlier and makes an escape.
Quite a few of these show up in Xenoblade, and they'll end if everyone gets KO'd, a certain amount of time passes, or if you cause enough damage to the boss. In the latter two situations, the boss will simply send everyone flying with a special attack to end the fight. Their HP also simply stops decreasing once it reaches the "You lose" threshold, so you can't win even on a New Game+, where you could realistically OHKO some of them otherwise.
A notable example pits your party against two bosses, with one 10-20 levels stronger than the other, and probably too strong for your party to defeat at the time, forcing you to choose the weaker boss instead. Defeating either boss moves the plot forward, but complicating matters is the fact that the weaker boss is Meyneth/Fiora, who is being controlled by Egil and doesn't want to fight you. On a New Game+ or with sufficient Level Grinding, however, you can wail on the stronger boss with no problems.
A truly interesting example that's impossible even on a New-Game Plus! Your first encounter with Morag in Magi-Nation begins like a normal boss fight, with Morag summoning one comparably weak Koil and doing nothing else until it rolls over. After that he gets mean and continually summons Borgors until they eventually slaughter you. Not only are the first few incredibly over leveled at this point, replacement Borgors (which you flat out will NOT see on your first playthrough because the first ones are so damn tough) are even higher in level, and Morag has 65 million energy, compared to the second-highest boss with roughly 1500 (who is not the final boss by the way). Later on you have a normal boss fight with Morag who is incredibly weaker, with only a few hundred energy, but has more variety to his monsters and also casts spells.
In Ys VI: The Ark of Napishtim, the secret room that contains the Eldian Orb, which is a Guide Dang It! to reach, is guarded by a innocent-looking goldfish-type miniboss called Majunun. Unless you really max out your experience(which takes forever, since in the higher levels all enemies only give 1 EXP), you have no chance at defeating it. In fact, its attacks will One-Hit Kill you at the level you are at when you can first reach the room.
And in the remake of Ys III, Oath in Felghana, the first battle with Nicholas Garland is unwinnable. His shield simply cannot be broken, no matter how much damage you pile on it. The only option is losing. Of course, the second fight is perfectly winnable.
Also, in Ys IV: The Dawn of Ys, Keith Fact is possessed and turned against you, and the only way to proceed is to be beaten.
Rentar-Ihrno in Spiderweb Software's Avernum 3 cannot be defeated in the final boss battle. This is not so much a hopeless battle, however, as the objective is to disable their doomsday machine.
Players who've managed to handle her with Wall of Blades spam report that nothing happens when she falls. You just go and turn off the machine anyways, and the ending progresses as normal. At least Spiderweb learned their lesson for the fourth game—the Shades regenerate to full health every turn, making their defeat mathematically impossible without the necessary weapon.
3, 4, 5, and Q all feature special bonus bosses (the Velvet Room siblings) who are usually just very, very hard to beat... unless you make the mistake of equipping the Omnipotent Orb, which blocks all attacks save Almighty-element attacks. Doing this instantly turns the fight in a hopeless one because they will spam a 9,999 damage almighty attack. Your health cap is 999. There are ways to survive a few turns, but there's no legitimate way to do enough damage to beat the bosses in that time limit.
In Persona 4, the fight with Shadow Rise mixes this with Bait-and-Switch Boss. Halfway through the fight, Shadow Rise begins to "scan" your party, becoming immune to all your attacks in the process. Teddie unleashes a Desperation Attack to save you, leading to Rise acquiring her Persona. It then becomes apparent that the actual boss in this dungeon is really Shadow Teddie.
The final boss of Persona 4 pulls this twice. Ironically, your 11th Hour Superpower then turns you into one for them. A harsher example is the final boss of Persona 3: even though you become immune to the One-Hit Kill, the only way to stop Nyx is to sacrifice yourself. To be fair, you technically win...
In Persona Q there are the FOEs. There are overpowered shadows found in every dungeon that can't be killed when you first meet them, so you have to run away from them (Mission Control suggests this at the start of every FOE fight). Anyway, by the end of each dungeon you should be strong enough to even kill the FOEs of this dungeon, but they're still tough enemies. Also, if you tackle the game on Risky difficulty, FOE battles cannot be escaped from. Ambushed by one and not leveled high enough to beat it? Better reload...
The initial fight against Shadow Nejima in Mementos, as your attacks will always miss him. However you must trigger this battle to start the Tower Confidant and use a skill received from said confidant to defeat him.
The first fight against The Holy Grail is unwinnable, as it will constantly heal itself.
In the finale, you get a preview of the Bonus Boss fight against Caroline and Justine. Unfortunately, they'll automatically reduce you to 1 hp in two turns, ending the battle.
Somewhere between Hopeless Boss Fight and Heads I Win, Tails You Lose, but closer to this one: The first battles with the Gizoid Centurions and Prefect Charyb in Chapter 10 of Sonic Chronicles: The Dark Brotherhood. The party is split up into two, and after one team starts to take on these almost invincible bosses, you get a cut scene of losing after three turns of survival (losing in the first three turns gets you a standard Game Over). After this, the action cuts back to the other team, which does something to neutralize the enemy's Nigh-Invulnerability shields. You want to focus on restoring your own health and PP, though, because when the battle resumes, you'll be at whatever status you were at the end of the previous battle.
And very soon thereafter, you have the first encounter with Fox Face. And all subsequent encounters, really: Fox Face is unstoppable, and you have to run away from him. Even when you do finally manage to defeat Fox Face, he's replaced with the four masks of malice, who are likewise unstoppable until a certain point.
This happens several times in Wild ARMs 5. The game doesn't even pretend to be fair during these battles: any attacks against your opponent automatically fail (with an explict "No Effect!" message) and analysing the enemy reveals that you gain no experience points for a victory.
While the first 'unbeatable' bosses are impossible to defeat, the rest of them can be beaten as long as you're not a one-man party, have a lot of revive items, and most of all, the ability to unleash finest arts (equipping a punching glove on a character with the Sword medium equipped, thus activating finest arts whenever he uses 'Sonic Vision'). Luck helps, too. Furthermore, you do gain a lot of experience points from the victory. Some examples of these fights are the first Golem the human characters fight or the first battle with the Ice Queen. However, the following cutscenes remain unchanged.
A variation occurs in Valkyria Chronicles on the first two occasions where you fight Selvaria. In the first case she's utterly indestructible, but if you just ignore her (and take loads of cover from her gatling laser) you can still complete the objective, and in the second she appears a long way from the main objective but spends her time casually blowing up your tanks and causing instant mission failure.
That second one doesn't really count - all you need to do is pop the Edelweiss out of a hole, hit her with a mortar shell, and duck the tank back into the hole. In practice, the only thing this does is to cancel out the extra moves you get by having the tanks on the field in the first place. However, if any infantry units are in the wrong place when she shows up, it rapidly becomes hopeless for them.
In a reversal, Alicia takes on Valkyria powers eventually and becomes this for the imperials, utterly ignoring gunfire while running around smashing everything she comes across
In Valkyria Chronicles III, the final fight against Dahau it's impossible to even scratch him, something he attributes to 'the force of destiny' (a pretty big theme amongst the Darscssen, including two of your party members). The real objective of the mission is to destroy the ancient superweapon he's going to use to set the world on fire.
In Radiata Stories, the first battle (against Ridley Silverlake) is deliberately rigged for you to lose. Your attacks are woefully underpowered, while she can deal double digit damage per attack. Even if you manage to avoid getting your HP axed to zero after about one minute, Ridley will pull off her Wild Pitch special attack that instantly knocks you out, no matter how high your HP is.
In Pokémon Ranger: Shadows of Almia, the fight against Darkrai is like this until your Capture Styler gets upgraded.
Pokémon Black and White and Pokémon X and Y have variants of this. The battle is hopeless for the boss; fainting it, it fainting all your mons, or running out of balls would just delay the capture of the mascot legendary.
The fangame Pokémon Reborn plays this straight twice: first, against the supposed leader of Team Meteor atop Pyrous Mountain, and then during the fight against Eland his Ditto, which has turned into an Arceus. Technically subverted should you side with Randomus for the second one, as if you do you have to win the battle to progress through the story. Actually beating either battle seems to lead to a better ending, though.
The fanmade browser game Pokémon Eclipse has these starting from Story 2 in the Battle Arena mode. This is indicated by the level of the opponent being at an infinity sign. The opponents range from Darkrai (until you kick the ass of Shiny Regigigas, which was the source of Darkrai's power, Story 3's Big Bad's Cresselia (with the Big Bad of Story 3 even mentioning it being fun to easily kick your ass), Celebi (who still praises your Pokémon for being strong and helps you), and Mewtwo in the Special Story Mode, which stops Mudkip's initial attempt at world domination.
In the PC version of the famous D&D adventure Temple of Elemental Evil, an evil god by the name of Iuz will suddenly join a battle which your party would've (by this stage of the game) been able to win reasonably well. Iuz has a VERY large number of hit points, and can easily kill a healthy barbarian in a couple of strikes. The idea is to try to avoid getting your entire team slaughtered (which can happen very quickly), until a couple of turns later another god shows up. At this point, everyone in the team is resurrected, as are the enemies, the two gods disappear, and the whole fight starts over from scratch. In other words, you're expected to try to survive for as long as you can. Surprisingly, there are actually people who've defeated the god, and the designers even added an extra "achievement" reward which is acknowledged in the endgame slideshow. Good luck though, it is extremely difficult.
In Lost Odyssey, the end of Disc 2 pits the team against Councilor Gongara. After he taunts them for several turns about how pathetically weak and ignorant they are, he wipes them out with a single spell.
In Digital Devil Saga, there's the Jailer, a Horse like demon that is scary as hell. until you weaken him by replacing his normal supply of canned fifteen year old with rotten meat. He'll use Black Bind which renders the party immobile and ends the battle.
Azoth of Destiny also has the first encounter with the Dragon King, Ardgevald. To say that Ardgevald annihilates you is a massive understatement: he unloads three immensely powerful attacks before you even get to think about moving, the third one doing overkill damage just in case the first two didn't beat you all the way to death yet, and the battle ends right there and then with your party flattened on the ground. You later on get to fight him for real after getting stronger, and while the battle is winnable, he still does enormous damage and can very easily curbstomp you again, showing that his earlier stomping of your carcass wasn't all plot power, but him being legitimately tough.
Near the end of Fossil Fighters you have to face the police chief who's actually the boss of the criminal organization you've been facing. His dinosaur that he summons has max evasion so every attack you throw at it will miss. Though it isn't superpowered the fact that you can't hit it makes it a Hopeless Boss Fight.
In The Elder Scrolls IV: Oblivion, it's not technically a boss fight but one of the optional daedra prince quests requires you to anger a widower and let him attack you with a special blade. The only way to win the quest is to die.
AdventureQuest has a few, such as The'Galin in "The Final Battle!" and the Nightmare Queen in "Frostval in July!". The first boss is literally the manifestation of a god who is Just Toying with Them. The latter is nigh-unbeatable given that the player character is fighting against a nightmare.
DragonFable's Fire War kicks off with a Hopeless Boss Fight against Akriloth, which sets off the Start of Darkness for Drakonnan. Some battles against Drakath also qualify, as does Xan, Sepulchure, and several more.
Also in DragonFable, it is possible to beat the Elemental Lords as Nythera rather than as Warlic. You get quite a chunk of experience, but the battle is retconned and you have to do it over. Strangely, the battle is a lot easier as Nythera.
In the DragonFable minigame Archknight, there's a boss you meet at level six or seven who has more than thrice your HP. Naturally, it's this trope and critical to the plot.
Two in Last Scenario. One, the first fight against Felgorn, is a true Hopeless Boss Fight, as as soon as you get him down to half his hit points, he hits you with a Total Party Kill attack. The second is more to show The Hero's Determinatorism than anything, though, since he attacks Helio by himself, with none of his equipment, and immediately afterward you get to really beat the guy up.
There is also a third that invokes this in Melchior, who fights you to teach your party a lesson. He is level 99 and deliberately designed to be nigh-impossible to defeat. You're supposed to escape from him, as it's meant to be a lesson in humility.
Though your party faces Nene several times over the course of Blue Dragon, the only time you can actually engage him in a winnable battle is the very last time you meet him, with all your other confrontations falling into this category.
One of the most annoying examples is The Legend of Dragoon. In the arena scene you have to fight Lloyd, a major enemy of the game. He is undefeatable in the fact that he will simply dodge every attack you throw at him, and then, to add insult to non-injury, he mocks you with such phrases as "Ha! Not even close!" or "Don't waste your time!" or "Too Slow!"
One of the first few things you witness in .hack//G.U. is the cutscene version of a Hopeless Boss Fight, Haseo vs. Azure Kite Tri-Edge. Haseo is Perfect Play AIed so thoroughly and easily it is almost laughable. A variation of the same scene occurs at the end of the anime .hack//ROOTS since both series use that event as a buffer zone and it is no less brutal.
One of the .hack novels also features TheOne Sin, an event boss that many players of the game thought was Hopeless. It turned out to be a particularly intricate Puzzle Boss which doubles as an egregious case of That One Boss (probably to the point where if TV Tropes exists in the .hack verse and some of the players were Tropers, they'd probably think that it's an exception to the Bonus Boss rule for That One Boss!), and the two players who defeated it (Orca and Balmung) gained extreme notoriety.
Additionally, the 4th game features the final battle with Cubia. Once you win, it uses a move called Sephira Returner to restore itself to full health. Kite and Black Rose immediately conclude that the battle is hopeless. What they arguably lack in genre savvyness, they make up for in innovation, as obviously the game didn't end there. In case you were wondering Cubia is connected to Kite's bracelet, once Kite destroys it, Cubia vanishes with it.
besides the cutscenes, several boss fights in the Demon Palace Tournament cannot be beaten, such as the fight with Bordeaux and Alkaid. You could over level your character beyond the needed level for the moment and hit them one time reducing them halfway down their HP bar, and yet the cutscene will roll with it showing you getting your ass royally handed to you even if two seconds ago one blow from your opponent did no damage at all.
Digimon World 3 for the Playstation features Zambamon, a Digimon that is way too powerful for your own Digimon to defeat at the time Dyou face him. In fact, he's so strong that the first hit he lays on your Digimon, you are forced to flee. However, he is blocking the path to the next major city in the game, so you have to find a way to get him to move that doesn't require battling him anymore than the first time, which is required to advance the plot.
In the Amaterasu server, trying to force your way into Amaterasu City gets you into an unwinnable fight against Knightmon. To add insult to injury, it's virtually impossible to run away from the fight.
The Super Famicom game Dragon Ball Z: Legend of the Super Saiyan, Goku has to fight Ginyu. Like the DBZ story, Ginyu has to switch bodies with Goku in order to continue or you can just let Goku be defeated. The rest of the party has to defeat Ginyu.
Dragon Age: Origins has what appears to be a Hopeless Boss Fight after rescuing Anora from Arl Howe's estate in Denerim. You are confronted by Loghain's chosen knight, Ser Cauthrien, at the front door, and while after about 20 seconds of trying to fight her and her guards you think it's one of those times you have to just die, it's actually possible to win the battle if you're really good, and the story continues on as normal, except that Ser Cauthrien doesn't confront you before the Landsmeet, what with being dead and all. You get a sword for winning the battle, but you can also get this sword by killing Ser Cauthrien when she confronts you at the Landsmeet instead, which is much easier.
Winning this fight also lets you skip the 'Captured!' quest, where you either have to be rescued by your companions or break yourself (and possibly Alistair) out.
Though the final Mass Effect 2DLC, Arrival, doesn't feature any bosses per se, at one point, it pits Shepard (alone!) against five consecutive waves of frighteningly smart, increasingly powerful enemies in a room that has little suitable cover and no ammo refills except the ones you pick up from the enemies. You thought the Thresher Maw on Tuchanka was hard? Good luck getting the "Last Stand" Achievement. It soon proves that regardless of whether you are beaten to unconsciousness by the enemies or taken down in the cutscene following your "victory", you are taken prisoner and the villain's plan proceeds.
For the record, if you survive, the Reaper artifact "Object Rho" reaches full power and concusses Shepard into unconsciousness, whereupon s/he is sedated. (It doesn't stick.)
Two other DLC packs — "Stolen Memory" and "Lair of the Shadow Broker" — feature bosses that restore themselves to full strength whenever you get their shields down. In both cases, your squadmates will (eventually) rig up an environmental kill: Kasumi wall-crawls up to Hock's gunship and disables its shields while Liara brings down the ceiling of the Shadow Broker's lair, annihilating him with the biotic energy conduit contained there.
Mass Effect 3 has the first Kai Leng battle at the end of Priority: Thessia, which is similar to the Shadow Broker example above.
In Ar Tonelico 2: Melody of Metafalica, there is a boss fight with a guardian for the goddess, which also serves as the tutorial for Replakia. Despite using it at maximum power for as long as Cloche and Luca could possibly manage, one-shotting her guards and dealing millions of damage, she just won't lose that last hit point. In exchange, the boss will pull out a move that is nearly impossible to guard without seeing the attack at least once. Perfect Guards and your best healing magic can last for a while, but that attack will eventually hit hard enough to take everyone out.
This is actually used earlier in the game when you face three members of the Grand Bell one of them being Leglius, and all this happening after Cocona goes IPD. While you could possibly level grind high enough to beat these three, the fact is that you play as Croix and only Croix, have no backup or Revyteils meaning he cannot guard or be aided by song magic, cannot harmonize to use better skills, and will die quickly if not right away due to the level you should be at at this point.
The first encounter with Bradley in Dubloon. Considering he attacks you before you can even prepare your attack, he could just as easily beat you up in a cutscenenote Not even to mention his attack may do less damage than your hit point max..
In the Costume Quest DLC Grubbins on Ice, the game opens with you fighting three characters of a significantly higher level than you without a healer in your party. However, given some skill in hitting the button prompts and luck and the enemy's choice in targets, you can win. That said, the game assumes you lost anyway.
In Vampire: The Masquerade – Bloodlines the default solution to the werewolf level is to simply outrun the thing until the timer runs down. But technically you can defeat it by getting it to crush itself in the observatory doors, but this approach is difficult to pull off even if you've been consulting walkthroughs.
Demon's Souls partially-averts this with a seeminglyunbeatabletutorial level boss. If you actually manage to beat him, you are teleported to an area where you meet a boss from much later in the game, the Dragon God. He kills you in a cutscene, just to be certain you die this time.
In Dark Souls, Demon's Souls Spiritual Successor, the first encounter with Seath the Scaleless is one of these. You die over time while you can't inflict any damage upon him. When respawning, you discover that you are in a prison cell, rather than the last check point, which is how you get to the fight it's possible to beat him at. While it appears to be an accident, you can in fact skip the first fight entirely by jumping onto the guard rail of the elevator to the fight while it's going down then jumping onto one of the ledges beside it, giving you access to the area normally only accessible by losing the fight.
In Hyperdimension Neptunia, the first boss fight with White Heart, Black Heart, and Green Heart cannot be beaten for two reasons. Not only are you fighting 3 very high level opponents who blast you each once with their best specials, but Purple Heart is controlled by the AI in this fight, guaranteeing you lose.
Similarly, in Hyperdimension Neptunia mk2 Nepgear, Compa and IF must face CFW Judge as the first boss, but that boss is so overpowered that he can One-Hit Kill the party and Nepgear's attacks won't damage it.
Hyperdimension Neptunia Re;Birth1 is a remake of the first game, unlike the first game, the example above is replaced by a cut-scene of the same battle, however that doesn't mean that the game is devoid of Hopeless Boss Fights, examples include Arfoire (identified as question marks for the first time. She accidentally Nerfs herself by copying Compa's power instead of Neptune's), Arfoire again (impersonating Green Heart, However Vert regains the HDD ability shortly after, placing Arfoire on even ground), The CPU Breaker (Which was only weakened by a Big Damn Heroes Moment from an unexpected character), and the aptly named CPU Killer (which precedes a a turning point and cannot be fought in a weakened state unlike the others). With a lot of Level Grinding these bosses can be beaten, but they give no experience, no credits and don't even appear in the game's bestiary, and the events still play out as if they won the fights.
In Winged Warrior III, the player is not supposed to be able to defeat the Nova Knight during the first encounter. It's possible to beat him after a lot of grinding at the training center, in which case the game will act up.
The Newgrounds flash game Mobile Weapon contains an example of this. At some point in the game, you fight Fleet Commander Ariel Highwater in her mecha, Serra Superion. The mecha has ridiculous health and power and will maim your party within a few hits. However, you don't lose the game: you merely need to repair your own mechas after the battle. You'll never hear from her again. The battle actually subverts the trope - if you grind a lot of monsters to level up to maximum level and stack up on repair items before facing her, she becomes the Bonus Boss instead. She will still be, by far, the hardest boss in the game. Defeating her will not alter the plot at all, as she will just run away from you. You will, however, get a great sense of accomplishment from beating a very hard boss. Oh, and the infinity plus one grenade launcher too.
Exit Fate: The first fight against the Demon Commandos. They're completely unbeatable and will curbstomp you with insanely powerful attacks. Several following encounters continue the theme - including one tactical battle, where they are both represented by sorcery units with a thousand manpower each, and a "victory" just means you managed to fend them off for long enough to escape. The two penultimate fights are also guaranteed defeats even though you face each of them alone. However, all damage dealt to them carry over for the final fight.
The Machina Vanguard from Baten Kaitos Origins are this once they hop into their fancy Machina Arma. You're free to fight them as many times as you like and although you can do scratch damage you only get so many turns until the game forces you to retreat. It's not until much much later in the game that you get an attack that can crack their armor and the game lets you fight them for real.
In Default the party encounters Victoria and Victor of the Council of Six fairly early in the story and though the battle certainly appears as to be a standard Asterisk wielder boss battle they're about to curb stomp the party entirely. Even if they don't the battle ends after five turns. Fortunately Edea makes the party (and the player) plenty aware that they're opponents that they can't possibly hope to beat as they are. However it would be unwise to just roll over and die and you should consider lasting long enough to try to steal from both of them, as their items increase Mind and Intelligence by 10, both of which is very impressive at that point in the game. It is possible to defeat this battle, either in New Game+ or with enough Socialization Bonus attacks of 99,999. The enemy will simply jump to "as if you had barely survived five rounds."
Bravely Second, in the vein of Final Fantasy II above, starts off with one of these where Kaiser Oblivion receives 0 damage from your attacks and inflicts 9999 damage with all of his. As the fight is scripted, you can't hope to win this one even on New Game+. Except not if you start after the end of Chapter IV, where the scripting only allows Yew to participate; if you attack normally, you still lose and get kicked back to the Prologue. Use Bravely Second, however, and the fight changes into something more winnable and allows you to carry on with the story.
Happens at least twice in a row in Three the Hard Way, against Benson and Mauldin. In the first encounter, Benson pulls a Death Technique that almost killed the protagonist, and in the second encounter, Mauldin uses a Sleep spell on the party before escaping. Unlike most examples, however, the story will only progress when they defeat you with the aforementioned technique. If they defeat you normally, you will get a game over.
From Shin Megami Tensei IV, there is Xi Wangmu in Ikebukuro, who subverts this trope. At first, all of your attacks do only 1 HP of damage, even Critical Hits and exploiting her elemental weaknesses. Only after the Ring of Gaea intervenes, gets curbstomped, and makes a final stand from inside her stomach does she finally feel pain and lose her steel-hard defenses, leaving her open to take actual damage.
Spandex Force, being a superhero parody, takes it to the extreme with a level 100 clone of your character which has 9999 hit points and a single attack called Ultimate Blitz, the description of which states simply "DON'T GET HIT!"
In A Blurred Line, you very first battle can be this, if you choose to run to work instead of waiting for the Hover-Tram and if you don't learn that Arden is a double agent. You're pitted against four Lazer Teeth thugs and no matter what equipment you buy at the shops with your 100 credits, it’s impossible to put down more than two before you’re defeated and taken prisoner.
Then, there’s the battle against Kersh at the Memnosyne’s house. He has quite a few powerful abilities, but can still be fought valiantly ... up until the 10th turn or so, when Talan ends the battle himself, saying that he’s too strong.
In The Witcher 2: Assassins of Kings the player must complete a regular boss fight versus two assassins, which ends once one of them killed. In order to get information from the dead man the player must then participate in a necromantic ritual that allows him them relive the assassin's memories. The memory plays all the way out to the same boss fight, but from the point of view of the dead assassin. Since this result of the fight is a Foregone Conclusion the player is only allow to do Scratch Damage in order to make the battle hopeless. If the player does somehow manage to deal significant damage then the memory simply ends early.
Arc Rise Fantasia has a few of these: the first fight against the leader of the QED, the fight against the rival team in Diamant Castle later on in the game (which you can't lose but merely survive for a few turns; the boss would be asininely hard if you needed to win), and the first time you face the character who turns out to be the final bosses. There's also the first fight against the rival team on the beach; but that battle can be won or lost by the player, the story progresses either way.
In Atelier Ayesha: The Alchemist of Dusk, your party's encounter with Keithgriff at the ruins of Hallos Village ends with him wiping the floor with you and making off with the Plot Coupon you were charged with recovering. All attacks against him deal only Scratch Damage, and his attacks are strong enough to one-shot anyone in your party. It's still possible (but ill-advised) to encounter enemies in your weakened state after Keith escapes.
Monster Hunter Tri and 3 Ultimate features the Lagiacrus, who is first encountered during a basic 1-star quest to gather Monster Guts. If you went off to do online quests to get some good gear, it's possible to break off some of its parts for some drops you're not supposed to get until later, but you cannot kill or capture it.
Another inversion comes in the other path. It's impossible to LOSE against the final boss of the Pacifist Route, Asriel Dreemurr, as much as he tries to convince you otherwise. No matter what Asriel does, or how many times you're brought down to 0 HP, Determination keeps you going.
A variation occurs in the Pacifist run with Undyne. She refuses to listen to your attempts to spare her, so all you can do is run away until she chases you into a Lethal Lava Land and passes out from the heat.
From a Neutral run perspective, Mettaton is completely invincible in his first two encounters and he isn't exactly "spared" so much as he leaves and runs away from your new yellow Soul Mode, respectively. In the third encounter, it turns out his taking damage from the Shooter Mode was only part of an act. Eventually, you have to trick him into turning around, then flip a switch to get him into the more vulnerable Mettaton EX. From here is a more "traditional" boss by the game's standards, where you can properly fight or spare him. His "Check" descriptions lampshade this:
His metal body renders him invulnerable to attack. His metal body STILL renders him invulnerable to attack. Seriously, his metal body is invulnerable!
Napstablook is a variation. The ghost takes damage when attacked, but once the player has beaten them, they will simply confess that they voluntarily lowered their HP to make the player feel happy. Since they're already a ghost and can't be attacked or killed to begin with.
The Dummy can't be attacked by the Player and mercilessly mocks them for that as well. Even their description notes how attacks are useless. Ironically, the Dummy attacks with magic missles and thosecan damage them. In the end, the Dummy gets chased away by Napstablook's magic tears.
Happens twice in the first few minutes of Alphadia:
The first battle of the game, much like Final Fantasy II, deals with some of the main characters being pursued by elite imperial soldiers. Exhausted and out of supplies, they go down quickly.
After saving the characters from the first battle, the hero and his brother face off against another group of soldiers. After dispatching them, they face their sergeant, who hits for max damage.
Rakenzarn Tales has a few, but you get fair warning in getting a message either to last a certain number of turns or an optional mission to deplete a certain amount of HP for bonus stats.
Lunarosse has some. They're usually indicated by the boss having the max possible HP instead of a reasonable amount for that point in the game.
In Trillion: God of Destruction, the fight against the titular boss in the prologue is impossible to win, even with the recently-killed Astaroth lending Zeabolos the remains of his power. Trillion does, after all, start out with 1 trillion HP, so your time throughout the game will be spent training the other Overlords to be strong enough to grind down its HP little by little and stand a fighting chance to save their underworld from almost-certain destruction.
In addition, the first battle against Mokujin (a wooden robot constructed to mimic Trillion's powers and moveset), is also impossible to win. After a set number of turns, it blasts you with Final Miasma, which kills you instantly. When you fight it again during the second cycle, it starts out with a more reasonable power level.
After this, Trillion is technically killable in all further engagements. Your first, second, and probably third and fourth Overlord choices are doomed anyway. Trillion is just that powerful, and killing it is not a one-person job. Your Overlords can only face it alone, and only retreat so many times. The inevitable happens, and the survivors have to keep trying.
The very first thing the happens after you get Trapped in Another World is a fight with Dondera Tank. You get to defeat Dondera Tank several times later in the story, and you do have your actual primary weapon at this point, but there's still nothing you can do here except get wiped out and watch Dondera Tank attempt a Finishing Move but then get blown up by someone else.
In Dragon Fin Soup, you can try to hold off the Hooded Man in the prologue, but he can't be killed. Once his health drops to 50%, he fully recharges and summons an endless supply of skeletons that get stronger and stronger as you defeat more of them.
You can't beat Sara in Chapter 2, you can't beat <C> in Chapter 4 and you definitely can't beat Victor in Chapter 5. Then, because the game wouldn't be complete without one more you can't beat Scarlet's Spiegel while on foot. All of these are justified by the plot.
Fairly early on in Cold Steel II, you're tossed into a boss battle with two members of Fie's former jaeger corps, Zephyr. Both have around 90,000 HP and it's said before the battle that the party has about a "20% chance" of victory. You're allowed to whittle around a quarter of that before both use their S-Crafts. The battle ends automatically a few turns later and it's revealed that they were holding back.
Battle Hunter has the monster Gon, which shows up once the deck of cards has been depleted. He has much higher stats than any other monster in the game, usually being able to kill most any character in two attacks at most. Even worse, if he kills the player who is currently holding the target item, it results in an immediate Game Over. Even if you do manage to kill him, he just respawns on the next turn, ready for a new round of ass-kicking while you're still recovering from the last fight. Simply put, him showing up is intended to be your cue to wrap things up ASAP.
The second Spectrobes game, Beyond The Portals, wants you to lose the first fight gainst the Dark Spectrobes. They have a lot more HP, ATK and DEF than your Spectrobes, but just in case that's not enough their HP can't go below 1.
The Game MakerUnlimited Adventures allows you to make one of these by providing a "Party Never Dies" option for combats, meaning that losing the combat doesn't end the game and all player characters are restored to 1 HP afterwards. Actually, there's no option to check whether or not the party has indeed lost the fight, meaning that the designer should ensure that the enemies are indeed strong enough to wipe the floor with the party.