Sometimes the plot demands that you fail. The storyline requires the defeat of the main characters in order to make a point or explain a key event, regardless of whether or not the player would allow their party to fall in battle. There are two ways to accomplish this: One is the Cutscene Boss, where the player has no control over the battle's outcome at all — and the other is the Hopeless Boss Fight.
This is a boss with Nigh-Invulnerability, if not sheer invulnerability outright. Odds are good you won't be able to damage or hit him at all — he'll quickly reduce it to Scratch Damage or just ignore it outright as he launches One-Hit Kill (if not Total Party Kill!) attacks on your team, and if you aren't able to run from this battle on turn one, it's Game Over on turn two. Or is it?
These bosses tend to make their appearance near the beginning of the game, before the characters have had a chance to earn Experience Points or Level Up, or otherwise become more powerful than their predefined starting levels — this makes narrative sense as well, because this is when the characters would be least likely to survive an encounter with them anyway. Done well, this defeat will strike fear into the heart of the player, having learned firsthand just how powerful this boss really is; done poorly, it feels like a cheap trick designed to advance the plot, Suspension of Disbelief be damned.
Since games like to pretend to be fair, your opponent in the Hopeless Boss Fight tends to return later in the game for a proper battle. They have the odd tendency to be Climax Bosses: If they were the Big Bad or The Dragon, players will get to fight them after this Final Boss Preview by the end of the game as a straight up Final Boss (or Penultimate Boss).
Ideally, the game should make it somewhat obvious to the casual gamer the fight is probably intended to be hopeless, lest you waste your serious healing items and abilities. This is another reason why these tend to appear at the beginning of the game, as you haven't even acquired any serious healing items or abilities yet.
Often, Genre Savvy players may confuse a hopelessly difficult boss with this and give up fighting on the assumption that the battle is intended to be hopeless ... only to discover it isn't. More fiendishly, some games can silently observe the player's efforts to determine whether they put up a decent struggle before going down, with a genuineGame Over issued if they died too quickly, or make the objective to survive for a fixed amount of time, with any deaths resulting in game over.
Not all Hopeless Boss Fights exist to defeat you at a specific point to advance the plot; there may also be Border Patrol, confronting players who try to go Off the Rails in certain areas with a Hopeless Boss Fight, from which the only option is to Run or Die. If the monster will become defeatable as you gain power, it's a Beef Gate.
Sometimes, the apparently invincible boss can technically be defeated, if you have a precisely tuned tactical setup, access to a Disc One Nuke, the advantage of a New Game+ or endless hours of Level Grinding behind you. In this special case, defeating the boss may net you a cool reward of Experience Points or rare items. Or cause crashes and glitches if the devs weren't expecting it.
In any case, since a Hopeless Boss Fight is one the game expects you to lose, the game will react to its defeat in one of several ways:
The Battle Didn't Count, and events unfold exactly the same as if you had lost the battle. Lazy scripting might simply assume that the battle ended with the boss wiping the floor with the party. In extremely rare cases the game may remember this outcome anyway, possibly as a factor for its Best Ending.
In Castlevania: Order of Ecclesia, at the final battle with Dracula, once you've whittled his HP down to 0, if you're not fast enough to quickly equip the Dominus Union he unleashes an unbeatable move that's supposed to instantly kill you (it does 9946 damage, though if you use the Volaticus Glyph, you can fly in the corner of the screen to dodge it, though the frame of Dracula and the explosion stays unmoving and never ends).
In Castlevania: Lament of Innocence Walter counts until the moment you hit him with Vampire Killer. If you don't and try to hit him with any other whip or attack instead, it'll be to no effect whatsoever. He CAN hit back though so unless you switch to the VK, you'll eventually end up dead.
In American McGee's Alice, Alice cannot defeat the Jabberwock the first time she fights him. The objective of the battle is survive and stall until the Gryphon can arrive to chase him away. (Succeed, and you get the final component of the Jabberwock Eye Staff, a powerful weapon.)
The truly ironic thing about that is, while the second battle with him doesn't fit the Trope (not only can she defeat him, she has to, because he'll kill her if she can't) the second fight is much, much harder than the first; probably the biggest reason is, he has enough room to fully utilize his ability to fly in the second battle.
Shadow of the Colossus has an unusual take on this: At the end of the game you turn into the final boss, you're effectively invincible and can attack the armed group that just tried to kill you. This is hopeless, though. They'll seal you away no matter what you do.
killer7 brings two semi-examples to the table, the first being the fight against the Handsome Men (essentially a series of duels, some of which are impossible to win, and in the long run you only win thanks to one character's Eleventh Hour Superpower) and the other being the fight with Greg Nightmare (who sends a group of near-invincible Heaven Smiles at you, who systematically take down six of your personas, leaving Garcian to pick up the Golden Gun and finish Greg off).
Oddjob from James Bond 007 for the Gameboy could not be beat without a special item, ensuring Bond gets dumped in the desert where he can't possibly survive.
Brave Fencer Musashi has the fight with the super-vambees in the church. They can't be killed by damage, and if they grab you it's instant death. Fortunately, if you hold out long enough, the sunlight kills them. There's also a few instances where there's some invincible thing chasing you and all you can do is run - the stone head in the prologue chapter comes to mind, as does one of the forms of the Final Boss.
In Ys Origin, when played as Yunica, the first fight with Kishgal definitely counts.
In Ys Seven, there's the first fight with Scias in Altago Palace where you MUST lose after his HP is at half, since ALL attacks are BLOCKED (0 Damage) from then on ALONG WITH an attack that does 2k+ damage to you.
In Pokemon Rumble Blast, at the end of World 2, you get a Team Battle with one of these. Lampshaded with a Help Sign stating that "things might not always go your way, but sometimes is a good thing." Hurts your pride if you don't read the sign, but is considered a Critical Hit after winning the Charge Battle to reach this zone.
At the beginning of the game, the boss fight is against Zekrom, the Penultimate Boss fought right before Dark Rust. Because of the nature of the game, it would be possible to defeat it - except it destroys the floor of the arena and flies away after either you or it take a set amount of damage.
In Pokémon Rumble, at the beginning, Rattata notices that the boss door is open and sneaks in. Of course, you can't win with level 28 against 100.
In Monster Hunter Tri and 3 Ultimate, the Deviljho basically functions as a hopeless boss fight capable of barging into non-hopeless boss fights-though it's technically killable, you have almost no hope of doing so(even being explicitly told this on your first encounter) until the very end of the high rank quests. In the meantime, you have to do almost every high rank quest up to that point with the fear of a nigh-invulnerable spiky tyrannosaurus with immense attack power barging in and stomping all over you, forcing you to either flee the area and wait for him and your target to split up or dung bomb him until he goes away.
Another is the first encounter with Lagiacrus, the game's flagship monster during the quest "Guts-It's what's for dinner." The player is actually supposed to avoid it, but even if they're strong and skilled enough to knock it's HP down to zero, Lagiacrus won't die. A rather odd case as due to the game's parts damage system the player can slice Lagiacus to ribbons and it will still keep fighting.
The beginning of Metro 2033 requires you to actually get killed by the amassing enemies, if you continue defending you will either get overran or you'll use up all your ammo.
Kratos's first encounter with Zeus in God of War II, where Kratos is tricked into draining his godly power into the Blade of Olympus, rendering him mortal. After the battle with the Colossus, Kratos is heavily weakened, barely able to swing his blades. At this point Zeus comes along and defeats Kratos.
Similarly, in Chains Of Olympus your first battle against Charon is hopeless, since you don't have the right weapon to fend off one of his attacks. You must cross all Tartarus to find it and get back to fight the boss.
Blood Omen: Legacy of Kain includes an early battle with Malek. He cannot be killed, forcing Kain to leave and seek help, first from the oracle (Moebius) and then Vorador, who defeats Malek himself.
In fact, the very first battle is against respawning bandits. If you kill them all, more will come along, and eventually kill you in order to progress the story.
In Dante's Inferno, you may perceive the first level's boss, Death, to be a hopeless fight (understandably so, seeing as he's Death), if you don't realize that his life bar decreases from right to left, rather than the standard left to right. In reality, he is a Wake-Up Call Boss, teaching you the importance of blocking.
This happens twice in Breakdown, both with The Dragon, Solus. The first time isn't much of a fight, since he slams you against a wall before you can really do much. The second time is more of a fight, but the second he gets more strength, the fight's basically over. However, you get to fight him again after some Time Travel, and after he remarks that you were supposed to die quick, you beat him because you're now as powerful as he is.
If you play the Zeon campaign in Mobile Suit Gundam: Federation vs. Zeon, nearly every encounter with Amuro and his Gundam is this. You're not even able to consistently damage him until the final mission, where you drop into his own final battle with Char.
Same with Char when playing the Federation side. For that matter, pretty much any named, voiced character from the original anime that isn't on your side and appears in more than one mission fulfills this role. Note that it is actually possible to defeat them before it's called for, it's just really difficult since they have hit points out the wazoo and they pretty consistently kill you in one or two hits, and the game will never make note if you manage it; as the campaign wears on and the end (or that character's time to die) draws closer, their HP continually lowers to more reasonable levels.
Eureka Seven has two videogames, the first of which has you get pit against a high level pilot before the tutorial runs. You are able to beat her with the correct amount of effort, but your first time playing you will be severely unprepared. If you do win, it doesn't matter anyways, it only changes the following cutscene a little and everyone still treats you like the newbie afterwards while Ruri is still hailed as the best.
Fullmetal Alchemist: Curse of the Crimson Elixir has the boss fight with Scar. Win or lose, the same exact cutscene happens where Ed and Al are defeated. If you win, you do get an item at least, but it doesn't change the story progression. Neither does the following battle with Armstrong, but that doesn't affect anything but the direct moment anyways.
The very end of the game, after you've beaten the final boss, pits you in a hopeless boss fight against both Mustang and Armstrong at the same time. You can attack and dodge them all you want, but they can't be defeated. You can knock them out, but they'll just get back up again a few moments later. You're intentionally suppose to lose.
Custom Robo Arena: The second time you fight Hadron, he is totally invincible, as he has completely absorbed Scythe's energy. You have to lose to trigger the next scene. Also, in a dream sequence at the very start of the game, you fight Jameson of all Robos. Technically, it's possible to win, but the dream ends a short while after the battle starts anyway.
The first boss battle of Metal Gear Rising: Revengeance has Raiden facing off against "Jetstream" Sam. Sam does fairly devastating damage, even when the player blocks his attack and can also Flash Step away. Also, in the first and second phase of the final battle Raiden can only do Scratch Damage to Armstrong at the best.
In the beginning of DNA Dark Native Apostle, while making your way though the first few rooms of the game, you encounter a boss, antagonist Greado. Even though you can hit him with bombs, he won't take any damage, so you end up getting defeated. After this, you get dragged away by security personal and dumped into the sewers, leading to No. 13's amnesia and thus the main events of the game.
Episode 6 of Asura's Wrath is this from the from story perspective, although it still plays like any other chapter gameplaywise.
Escape from Monkey Island ends with a Boss Battle that could go on indefinitely, as the players regenerate their hitpoints faster than the other can take them away. Naturally, there's a trick to winning this one.
In chapter 2 of Imprisoned, you have to kill two guards in a row, and then the third fight is against a machine, which you can't damage. Later on, with a special ring (and/or a whole lot of Level Grinding), you can beat them. The fight with Jade is similar in that you can't damage him because he needs to beat you for the game to continue.
The flash game Armed With Wings 3 has a battle with the evil king Vandeer Lorde as the third boss, which you're plotwise not powerful enough to kill by then. He gives you 30 seconds to live, so the boss battle ends after those 30 seconds with a cutscene, and the goal is to survive. A little annoying since he's not all THAT tough, and it would be possible to actually kill him, especially if you use a New Game+. Makes it all the more satisfying to kick his ass when he returns as the final boss, in pretty much the same state.
At one point in Cosmic Spacehead you find yourself in an asteroid field and are told to keep dodging the incoming rocks. The action sequence continues until you take too much damage, at which point you are forced to take a scripted stop at a space station. Unfortunately it's quite easy to dodge every asteroid that comes your way, which means that a good player may end up continuing the sequence indefinitely, or at least until they get bored and decide to throw themselves into a rock on purpose.
Actually present at the end of Quest for Glory III. As the Hero and his newly-arrived allies make their way into the Lost City in search of the World Gate allowing Demons access to the world of Glorianna, they must each fight an Evil Twin of themselves. It doesn't matter how high your character's stats are, what abilities or spells you have, or how good you are with the combat system, the player cannot defeat his doppelganger. As the fight drags on and the Hero is close to being defeated, Harami, who at first refused to fight, suddenly appears and Backstabs the Hero's double, distracting it long enough for the Hero to withdraw from the fight and head for the final confrontation with the Big Bad.
Card Battle Game
Etherlords II has a couple of these, most notably the "Dragon Musician" which has a level higher than any other thing in the game, including the final boss, who is basically the god of the universe, not to mention the crazy buffs he gets for free at the start of every fight(one wonders why the Vitals didn't conquer the rest of the world with this crazy powerfull being).
Though it is possible to kill him, albeit being VERY difficult and breaking one quest which was tied to him, if you manage to kill him, you basically won the campaign, because the rest of your poor enemies no longer stand a chance against your highest level character.
A lower level example of this can also be found in the Synteth campaign, where you have to summon a monster to aid you n invading a fort, because the enemies there are of too high of a level, but the thing is you can kill them yourself if you are a good player and have some luck, but it will reset to before the fight happened as if you lost.
In the DS Dinosaur King game, one of these is fought against Seth - he counters every move you can make regardless of what you do. A variation occurs in that it occurs after a normal boss fight with him.
Somewhat unusually for a Yu-Gi-Oh! game, Yu-Gi-Oh! Forbidden Memories has one of these, the first time you encounter Heishin. If you do manage to beat him, he simply challenges you again. (And again, and again, and again, if need be...)
Another example was in Yu Gi Oh! Nightmare Troubadour, where you had to duel against a Brainwashed Jonouchi. This was an odd case; you could win the duel, but doing so would result in a Game Over, as would losing. The only way to gain a positive outcome was to end it in a draw. (Yugi gave you a copy of Self-Destruct Button and Ring of Destruction before the duel occurred, both of which would help do that, if you managed to use them right.)
In Monster Monpiece, you get to fight the Masked Diva after her identity is revealed. She has 9 starting health (the max anyone else has, including the Bonus Boss, is four), 90 starting mana, on top of using a high upkeep Fusion deck (which 90 mana will let her easily spam from turn 1). If you can somehow survive the initial ten or so cards, her regen is the standard 3; but there is an immensely small chance you'll break even one of her cards before you're overrun the first time through.
In Monster Racers, you face off against world champion Reinhart in Ayers Rock shortly after winning the Star Cup. He has a level 35 Furion, whose stats are far and away higher than any of your monsters at this point, and it will outrun you effortlessly. You won't lose any Will for losing the race, and Reinhart will praise your effort in spite of your loss.
In one case for Need For Speed: Carbon, you can't ever get away from your pursuer in the very first race.
A very bizarre example exists in the Megadrive/Genesis port of arcade game Super Monaco GP. G. Ceara, who is in no waysupposed to belate F1 legend Ayrton Senna is supposed to be set up this way. He appears during your second racing season and is supposed to defeat you in the first few races no matter how fast you drive until you lose your contract to drive the best car in the game. Upon doing that, he will then become beatable normally for you to earn back your position with the top team. However, the way he's coded, if you can drive a string of perfect races early on, you can defeat him and bypass this entire scenario. Nothing changes except that you don't change teams.
Racing Lagoon, a battle against Kyoji Nanba's monster RX-7 is likely to be unbeatble. It's possible to win with a lot of turbos, but the result's not different.
In the arcade version of the Fighting GameRival Schools, you start off fighting students from other schools, attempting to uncover the mystery behind a rash of student disappearances. But in the third round, Raizo, the principal of Justice High School and Batsu's Disappeared Dad, shows up with a brainwashed person (his partner for the 2-man special attacks) to fight you. True to the trope, Raizo is very powerful and extremely resistant to damage at this point, but with enough skill, you can manage to chip off enough damage from him and fend him off until time runs out. If you lose as the plot demands, you continue to fight other students and unravel more of the story. If you are playing as the two Justice High teachers and lose, you still continue to fight students... as a Brainwashed and Crazy agent for Raizo, complete with color-changed outfits, until the plot has your adversaries fight you to snap you out of it. On the other hand, if you do beat Raizo here, you skip directly to the end of the game, with another fight against him (depowered to beatable levels, of course). Either way, once you defeat him you then fight The Man Behind the Man, an Ax-Crazy, sword-wielding Justice High student named Hyo.
Slightly different with the Gedo High in which Akira fights with her teammates instead. However, the said opposing teammates still take less damage than usual. They are quite beatable though if you actually take the time and are careful enough not to get hit too much.
The sequel Project Justice gives you a similar setting in Story Mode. Only the opponent is "Vatsu", an empowered Batsu who is actually Kurow Kirishima aka the local Smug Snake, only under a disguise. He reveals his identity once the fight's over, and then you have to fight him. Also, depending on the team you chose, he might have an extra incentive for your team: he's holding either Hinata (if you chose the Taiyo team that included her, Kyosuke and the actual Batsu) or Kyouko-sensei (if you chose the Teachers Team with her, Hideo and Hayato) as a hostage.
In several wrestling based games, there are matches where, even though they are hard, you can easily win. However, due to the storyline calling you to not have "come out on top" or not possess a certain title, a cutscene will play that takes the title/glory away from you. Several examples in Smackdown vs Raw 06 include: HHH losing in a steel cage, but Eric Bischoff taking away your title, defeating Eddie Guerro but the Undertaker interrupting the match, winning the tag team title, but having to give it to the injured wrestler you previously replaced.
The original SmackDown vs. Raw has a big example itself. At one point, you are given the option to join Vince McMahon, who will referee a WWE Championship match you're in. If you take his offer, it is impossible to lose as he will not recognize submissions or count past two when you're pinned. The inverse, however, is that if you turn him down the same goes for your opponent, meaning it's only a matter of when you lose.
You CAN win the fights, if you're extremely good at dodging, and very patient. But the cut scenes take place like it never happened, since the cut scenes are right out of the anime.
Dragon Ball Z: Budokai Tenkaichi 2 plays with this. Certain fights are normally impossible to win, and losing them advances the plot. However, level up your characters enough and you'll be able to alter the story's progression by winning. This opens up alternate story paths that shift around events and characters, the results of which are often hilarious.
Dragon Ball Z Hyperdimension for the SNES has fights in the story mode which you are intended to lose but with enough skill you can win these fights, however winning these fights has no effect on the story and the characters act as if you lost regardless.
Magical Girl Lyrical Nanoha A's Portable: The Gears of Destiny: The earlier battles against System U-D are like this, though they're beatable with enough skill. If you do manage to win, it will be revealed that the now exhausted character did little to no damage to System U-D, with the story continuing as normal after U-D retaliates. Having said that, beating her during these Hopeless Boss Fights is one of the prerequisites for accessing the Playable Epilogue and unlocking System U-D as a playable character.
In Duel Savior Destiny these get pulled on you more frequently as you get into the later story routes. For example, having to go up against four boss units at once all by yourself. During the last route, the game even dumps characters you've never had a chance to use or even have as your ally before, apparently expecting you to lose. If you don't, the game just undoes your victory anyway so you can win the way it wants you to.
The beginning of the ruined zoo in Super Smash Brothers Brawl has the Porky Statue from Mother 3, which cannot be defeated... well technically not. Without cheating, using Smash hacks, or somehow changing his absurdly massive HP, you can and will never manage to deal nearly enough damage to it to break it.
In Kamen Rider Battride War, Kuuga's Memory Level reenacts the final battle in his show. In fact, the entire level is one big boss fight, starting with a hopeless one. Every attack you preform against the boss is worthless and futile, up to and including finishing attacks. Then, after a cutscene where the boss sets Kuuga on fire, he transforms into Ultimate Form and you're finally able to fight him.
First Person Shooter
For the Tiny Tina's Assault on Dragon Keep expansion for Borderlands 2, Tina is the dungeon master for a roleplaying game called Bunkers & Badasses. Just as the Vault Hunters approached the gate to a hub world called Flamerock Refuge, Tina narrated that a dragon appeared out of nowhere to fight. It was impervious to weapons and its attack instantly sent every vault hunter into Fight For Your Life mode. When called out for this by Lilith, Tina revived everyone and substituted the dragon for a different, easier boss.
Deus Ex has a well-executed instance of this, where you are confronted by a small army who demand your surrender, led by Gunther Hermann (a main character). You have the option of resisting (even if this is obviously foolish), but Gunther is invincible in this fight, so there's no way to avoid getting captured even if you manage to outfight everyone else (which is pretty hard to do, so you will likely not even notice Gunther's invincibility). You get to fight Gunther later, when he's quite mortal.
It is quite possible to be starkly confronted with Gunther's invincible nature by taking advantage of his ridiculously stupid AI; lure Gunther into the train station and make an escape through the tunnels. Destroy the small army of UNATCO troops and bots awaiting you, and then find out (to many tropers' eternal frustration) that even though you can hear the evac chopper waiting for you, you can't get to it; you're forced to battle Hermann.
It is actually possible to get to da choppa by putting LAMs or gas grenades on one of the shack walls, and then climbing them. If done properly, this will get you past the invisible walls and invisible ceiling and you can get to Jock's copter. Even if you do this, he will still completely ignore you, forcing you to either kill yourself to continue or load from an earlier point.
If you trap Gunther in the train station as mentioned above, and manage to survive the other troops and bots outside for an extended amount of time, the game will move to the next plot point anyways just as if you had surrendered or been knocked out.
The climactic battle in the expansion pack to Jedi Knight: Dark Forces II, Mysteries of the Sith, presents a unique Hopeless Boss Fight when Mara Jade fights Kyle Katarn (now seduced by The Dark Side). Kyle's attacks are normal, but he is entirely invulnerable and hence impossible to defeat. This situation almost counts as a Puzzle Boss, except that the solution is not a test of wit.
Jedi Knight II: Jedi Outcast gives another example. Justified in that, alas, he has Force powers and you do not. The justification is even better than it sounds, because theoretically he has an unbeatable counter to any kind of attack you use. Blasters and energy weapons? Deflected by lightsaber. Thrown explosives? Force Pushed back. So it's not likely the player will even get the chance to verify he's invulnerable as well. Of course, if you happened to be less than skilled and used a cheat code to make yourself invincible, the game becomes unwinnable, since neither side can die.
In the last fight in Jedi Outcast you can reverse this trope on Desann (though it is hard). This is possible by trapping his lightsaber under the weak pillar in the room by making it fall on top of it when Desann uses a saber throw. He will then be unable to defend, and only use his force powers to force pull his saber, which is stuck.
Kinda like this in Jedi Knight: Jedi Academy. You have to defeat a mutated rancor. And when you do, the game respawns it immediately. If you do follow the (not so) logical level layout, you can get to the scripted ending where it dies for real.
The same goes for the level where you have to distract a rancor so the prisoners can escape, and you escape as well in the end. You can actually kill the rancor, but the game respawns it immediately.
When you first encounter the Makron in Quake IV, you can't beat it. You can die trying, however; unless you let the Makron catch you with its gravity beam, and thus capture you alive.
Sometimes the gravity beam glitches and kills you anyway (perhaps by wanting to drag you through scenery):
Inverted in the original Halo, there is a scene where several waves of enemies (but no boss enemy) are used to herd the character through a scripted scene. Perplexingly, however, the enemies aren't unlimited, and it is possible on the lower difficulty levels to beat them all, while Cortana keeps telling you that the only safe way out is to jump...
Played straight with the post-credits Last Stand mission in Halo: Reach, where you fight increasingly tough waves of enemies until Six is killed.
In Half-Life if you refuse the G-Man's offer at the end of the game, you materialize in the middle of a Vortigaunt building with an endless number of enemies...and no weapons. Additionally, both of the first two gargantuas encountered were apparently intended to only be beaten by luring them into a power reactor and an airstrike respectively, but through an oversight (likely to make it vulnerable to the airstrike), if you hit it with enough ordinary explosives you can kill one early.
While not technically a boss fight, a scripted event early in Half-Life 2 forces you to be overwhelmed by Combine forces before Alyx saves your sorry hide. It's impossible to skip this or defeat the Combine soldiers because you don't have access to weapons that early in the game. If you use a cheat to give yourself weapons early and kill the soldiers before they wail on you, the event doesn't trigger and you're stuck in a small corridor with nothing to do.
Midway through Half-Life 2: Episode 2, you encounter the acidic antlion queen in the antlion lair, who cannot be hurt until you're back outside later in the game.
According to the commentary, not only were you not supposed to fight the Guardian in this section, as a general rule you didn't have enough ammo to actually kill it. That didn't stop playtesters from trying, though, so they had to add a line from the Vortigaunt expressly forbidding you to kill it so that you wouldn't try.
In Call of Duty 4, during the end mission, You are told to shoot down a pursuing helicopter. Naturally, you can't.
It actually IS possible, with the aid of cheats. Doing so triggers a comedic hidden ending.
In Rise of the Triad, your first fight with El Oscuro cannot be won by simply shooting at him until he falls down. Firing at him will cause him to mirror your attack with a magical variant of his own. Just wait it out, avoid his attacks, and eventually he'll run away and trigger the next level.
The first level of the time-traveling FPS Darkest of Days has you as a member of General Custer's cavalry during the Battle of Little Bighorn. That goes about as well as you would expect.
Episode 1 of Doom ends this way (as does one level of The Plutonia Experiment if you choose the wrong one of its two exits); you get teleported to a dark room full of monsters, with a floor that not only drains your health but also turns off the god-mode cheat if in use, and you can't move; apart from shooting the monsters, all you can do is wait for your health to drop to below 11%, which ends the game.
In Battlefield 3, during the second and final mission as Miller, you have to hold out for an evac helicopter to come save you and comrades from your disabled tank with its .50 cal machine gun for 15 minutes. It's impossible.
In Far Cry 2, you start the game trying to flee a town while a civil war is going on all around you. You can't successfully escape the town, because even if you avoid getting killed by gunfire, you collapse from malaria before you can get far. Once you're out of commission, one of the factions picks up your semi-conscious body and drags you to safety to get you to work for them. Which makes very little sense, considering that you may have just shot a lot of them... and then either lost, or collapsed from your terminally serious case of malaria.
The very first Big Sister encountered in BioShock 2 cannot be defeated no matter what, though she will eventually flee if either her health or Delta's gets too low.
Anubis, the Final Boss of Zone of the Enders, is unbeatable, and that's it. While you can just wait for the end of the game to happen, if you are fast with the controls you can avoid any damage from him whatsoever, at least on easy. The only damage taken is that in a cutscene about halfway through the battle. What makes the battle that much more frustrating is that you spend the entire game (which isn't very long, mind you) fighting bosses that condescend and belittle you for being a kid, despite wiping the floor with them time and again. What do you learn when you reach Anubis? They were right. It is literally impossible to land a hit on the thing because it teleports from place to place constantly, away from you and dangerously close; but, to make matters worse, it can kill you if you're foolhardy enough to try to take it on (or are playing on the higher difficulty levels). The only option is to wait for the Atlantis to save you and run away.
Then he returns in the sequel and is unbeatable again the first time you fight him. The difference here being that it is actually possible to inflict some damage to him during the battle. Unfortunately, you're still too weak to take him on fully. It isn't until the very end of the game, after getting the same teleportation ability that it has, that you actually can win, and it is Satisfying .
In EverQuest Kerafyrm the Sleeper in the Scars of Velious expansion. He had insane HP and attack power for the time on top of the death touch ability. He was not intended to be defeated, instead utterly destroying your raid party and the inhabitants of Skyshrine, then disappearing from the game until he resurfaced as the killable Big Bad in Secrets of Faydwer.
That didn't stop the top three most powerful guilds on the Rallos Zek server from uniting together in 2003 and proving to the entire world that he was technically killable. Zerg Rush tactics worked out quite nicely. His corpse had no loot.
Final Fantasy XI has Absolute Virtue. When it was first released, it was supposedly killable ("the Development Team killed it with 18 people"), but anytime players have managed to kill it Square has nerfed the method that they used within weeks. Worse yet, their released information on how they killed it required violating their own TOS. For a long time, it remained this; however, Absolute Virtue is now defeated on a regular basis by many linkshells.
Another megaboss, Pandemonium Warden, also fit this category; in fact a story of a linkshell fighting it for several hours and finally submitting to fatigue and other personal needs became newsworthy when the boss was first unleashed.
The final mission of the Statesman's Task Force in City of Heroes features Lord Recluse, who's sucking the power from every hero in Paragon City, except you and your team, via four collection towers. He is completely impossible to defeat while the four towers remain standing. He can barely be hit, and if he is hit, he takes little damage, and he regenerates more health in one second than it would take a team of 8 damage dealers to do in a minute. And he can one-shot most tanker-type characters.
Also: (ISSUE 15 SPOILERS) In the new set of Task Forces, the Arch-Villain Riechsman takes no damage the first time you see him. You spend the next few missions devising a way to damage him, then put your plan into effect in the last mission. Even then, he's no push-over.
Kingdom of Loathing had Don Crimbo in the 2009 Crimbo Quest; after he inevitably beat you up, your character would give a speech that convinced him to give up on his evil schemes for the time being. There's also The Whole Kingdom, a Bonus Boss that cannot be defeated by any known means; it also has difficulty defeating you because it does almost no damage, but it eventually achieves Victory By Endurance.
One example that's still available is Cyrus the Virus. He's invincible and always defeats you after one round, but if you use specific items to make him stronger in 3 separate fights, he decides that he's too overpowered to hang around beating you up, so he leaves (after beating you up one last time). Afterward, a different area gets ravaged by a certain virus, which allows you to complete a quest there.
Mabinogi Fantasy Life: In G13 (Hamlet), you are instructed to fight the Grim Reaper, who is trying to kill Shakespeare. However, this is impossible due to the Grim Reaper's high immunity.
World of Warcraft has a variant in this in the Halls of Reflection. The boss is impossible to kill but the encounter is more about running away from him and killing waves of his minions fast enough before he catches up to you and kills everything in one hit.
Sort of repeated with Arthas in Icecrown Citadel. He instantly kills everyone when his health reaches a certain mark, but it's not over yet... King Terenas' spirit breaks free from Frostmourne and brings everyone back to life, while also weakening Arthas to the point of being a free kill.
The Vizunah Square mission in Guild Wars: Factions ends with Shiro Tagachi showing up and killing the entire party. It's not over yet, though, as the Envoys appear to resurrect you and explain the situation. This can be a problem, however, as the any necromancer minions (which generally prove very useful in the rest of the mission) have become unbound when the necromancer died and are now wailing away on the resurrected party while the leader chats with the envoys. Parties have been known to wipe this way.
Phantasy Star Universe has one of these in the first chapter of episode 2. The battle against Ethan and Liina is impossible as you cannot deal any damage at all. All you can do is endure long enough for a cutscene to take over.
The "Whisperdoom's Spawn" adventure in Dungeons & Dragons Online includes a section where you have to destroy the giant spider Whisperdoom's eggs while avoiding the spider herself. On normal difficulty, she's level 11 (the quest is level 6) and has massive damage reduction and a Healing Factor. It's best not to attack her directly and just smash the eggs.
Fisher-Diver has Captain Connell, who cannot be damaged. He eventually kills the player. You cannot escape him either as the oxygen supplies stop working when trying to return to surface.
Danganronpa technically has "culprits", not "bosses", but the trope plays out the same way in chapter 5. You cannot solve chapter 5's murder, at least not in chapter 5 itself. This is because only the player character, Makoto Naegi, and Kyouko Kirigiri are the suspects in that case. At one point Kirigiri tells a lie in court, but if you reveal it Kirigiri is executed and you get a bad ending. But not revealing the lie causes you to get executed...only to be saved at the very last second by an outside source. So chapter 5's "boss fight" is hopeless because you cannot win. You can only progress to chapter 6 if you lose in the correct manner.
Vile, the first boss of Mega Man X, is unbeatable, and in order to advance in the game, you have to let him beat the crap out of you. A similar situation happens later on in the 1st section of the final level, but Zero sacrifices himself so that you can fight Vile without having to deal with his indestructible giant mech.
With patience it is possible to survive the fight with Vile, but he still doesn't see you as a real threat and only leaves when Zero shows up.
When you are down to low amounts of health, Vile will try to jump away and shoot a paralyzer beam. If you're skilled enough, you can dodge the beam, run into Vile, and take enough damage to die. Take that, Capcom!
In the remake of this game, Maverick Hunter X, you can beat Vile very easily, and you are in fact required to do so (he'll kill you easily, no paralyzer beam this time), but he uses what basically amounts to a sucker punch to achieve the same situation in the original game. In the rematch, the Hopeless Boss Fight is skipped entirely and goes straight to the Heroic Sacrifice, which then leads to the mano-a-mano.
Similarly, in Mega Man & Bass, the first real fight with King is unwinnable, requiring Protoman to perform a Heroic Sacrifice, enabling you to actually hurt King. Like Zero, Protoman doesn't die for real. Even though he was cut in half in the opening minutes by an axe.
Your first fight with High Max in X6 is unbeatable; you just have to wait out his attacks until a cut scene triggers.
Given how the game was designed, any of the times you fight him with X after the first time can be unwinnable because it's possible to fight him before getting a way to damage him. Except the fight never ends after the first time and you die if you lose.
In Mega Man Unlimited, the final boss fight against a prototype version of Zero is hopeless. He deflects every weapon you fire at him, and the fight only ends when he takes you down to one unit of life and dislodges your arm cannon.
In Super Metroid, the Mother Brain begins as a hopeless boss fight, indestructible and using a wicked beam that pins Samus against the wall. After reducing her to 1HP and preparing to fire again, the baby metroid attacks it, draining it into helplessness, then feeds that power into Samus to restore her. The Mother Brain awakens and kills the metroid. An angry Samus (and likely an angry player) then opens the proverbial can with a new blaster weapon - the Mother Brain's super-beam, which smacks it around like a tetherball.
You still have to fight well enough to survive her uber-attack at least once, and you have to do enough damage to trigger the uber-attack (Mother Brain will not use it until she's taken a lot of damage). If you don't survive the first one, the cutscene will not happen and you just get a standard Game Over. If you do really well, you can survive this attack multiple times, but the battle remains hopeless until your energy has brought down low enough for the cutscene to take place.
Ridley at the beginning of Super Metroid is also nearly impossible to beat: You can take enough damage (until you are below 30 Energy, more specifically) or hit him 100 times, whichever comes first, since both events cause him to fly away. There's an amusing little twist on this one: if you do manage to hit him enough times, he will drop the Metroid briefly before picking it up, suggesting you at least hurt him.
Likewise, at the end of Metroid: Fusion, you face the Omega Metroid, and must let it claw you down to 1 HP, at which point the erstwhile Big Bad, SA-X, attempts to kill the Omega Metroid and is struck down with one blow, giving Samus the chance to absorb it and regain the Ice Beam.
A really strange version occurs in Revenge of Meta Knight, one of the games in Kirby Super Star. The first time you face down Heavy Mole Lobster on the ship, he actually isn't invincible to your attacks or strong enough to KO you with one hit. Your attacks are effective as normal. However, while you're fighting, Meta Knight is having a dialogue with his crew on the bottom of the screen. The dialogue doesn't last very long, and the second it ends, Kirby gets blown off the ship. So the player hypothetically could defeat the boss, but there isn't nearly enough time.
If you are really fast, skilled, dodge every attack without taking damage, never cease damaging Heavy Lobster, and have an appropriate weapon, it is just barely possible to kill it before you're blown away. Of course, because Heavy Lobster is only really meant to distract you at this point long enough for said blowing, it changes nothing.
In the Meta Knightmare Ultra mode of the remake, Kirby Super Star Ultra for the DS, you actually do have to beat the boss, both here and at the later point in the game where you'd face him for real as Kirby.
Drawn to Life: The Next Chapter's final battle with Wilfre must be initially lost to continue.
You Have to Burn the Rope subverts this trope. Your normal attacks can't harm the Grinning Colossus enough to even remotely get his healthbar down, but on the other hand once you burn the rope, you give that Colossus a one-hit kill.
The Legendary Starfy has one in the final world before you defeat Mashtooth for the first time. Starfy's Dragon form is useless against Mashtooth, becuase Mashtooth has a flippin' sheild! So the battle must be lost in order to continue, where you battle Mashtoth for real as normal Starfy. This does not happen after defeating Mashtooth for the first time, not even in the boss rush, for it skips to Normal-Starfy-Versus-Mashtooth fight.
The first fight with the Great Devil in Adventures of Lolo 3, 3/4 of the way through the game, is unwinnable. There's nothing really to indicate that you can't beat him, the only hint being that the manual says there are 3 sections of the game, while this fight takes place in the 2nd section. Not so bad, since there are no items for you to waste, but has the potential to be a nasty surprise if you thought it was the final boss, only to discover that there are still 25 more levels before that. Not to mention that his only attack is firing very easy-to-dodge projectiles, which means you could keep up the fight a long time without realizing it was useless.
Real Time Strategy
The second Bavakh Brother in Immortal Defense, who ends up destroying all life on your homeworld. It turns out it is possible to beat him, if you're crazy good and crazy lucky, but even if you do the game continues as though you hadn't.
Warcraft III involves enemy heroes with Divine armor, which can only be destroyed by units with Chaos damage. Because no ordinary units in those scenarios have Chaos damage, the bosses are invincible. Examples include Cenarius and Tichondrius. The player must seek a power-up to gain the Chaos damage required to kill the boss and win the scenario.
Then played dead straight in final mission of the original, which is basically a Hold the Line mission. You cannot hurt Archimonde (at least without using cheat) as you are only supposed to keep his army from reaching Nordrassil gate until the time runs out.
Incidentally he's easily killable with the godmode cheat built into the game engine itself. However, even after the destruction of Archimonde and all of his forces, the scenario still proceeds as normal (so if one of your ally bases gets destroyed, you will see the pre-programed cutscene of him gloating about it). However, all cutscenes will not have him anymore, as it utilizes his in-game model to do so. This ends up hilariously having a glowing circle on the ground gloating at you and causing all the destruction in the ending. The absolute final cutscene, however, is a pre-rendered CGI video, so Archimonde will appear as normal.
Similarly, one of the early missions in the Human campaign has you having to defend a small town against endless waves of undead. Given the small amount of resources, upgrades and units that you can use to set up a decent defense, you can never actually destroy the two undead bases that keep sending their spawns at you. By the end of the mission, not even a perfectly positioned series of towers and soldiers can stop the undead from overwhelming you. To make matters worse, there's a side-quest in which you have to prevent a THIRD undead army from being spawned.
In Pikmin 2, the Water Wraith. The game's Ninja Butterfly even tells you "Run Away! Run Away very Fast!" Any attacks against it result on the Pikmin being killed until the last sublevel of the dungeon when you recruit the purple Pikmin and can make him tangible.
In Kingdom Under Fire The Crusaders, during Gerald's campaign, you supposedly get one of these against Regnier. Your commanding officer tells you to flee the fight and protect the king. Its only supposedly hopeless because you can actually beat Regnier's unit by having your archers heal you and fire into the melee, while your infantry charge in, steal a few technique points (enough to heal) and then high-tail it back out. This allows a level 7 Gerald and level 9 Hugh to defeat a level 50 Regnier. You still have to leave the battle though, and it doesn't alter anything. (Though you do get a ton of experience.)
Happens later on while trying to sneak past a massive army of level 50-100 orc units. You're supposed to go around them. Sneaky players can lure them off one at a time and defeat them using the paladins you have with you to heal. The dark elves which try to nuke you with meteor spells are, however, invulnerable, and your only recourse is to dash past them. This leads to the incredibly frustrating situation where you've beaten around five units of ridiculously high level orcs and swarms upon swarms of weaker enemies if you went to wipe out every last enemy on the map, only to get blown into oblivion by some stripperiffic"vellie" mages. Gosh Dang It to Heck!! Again, you get gamebreaking amounts of experience and gold for this. Just as well, as the hardest battle in the game is just around the corner.
Similarly, in Ellen's campaign there's a mission in which you're supposed to just run past several armies of ecclesian knights and paladins. While you can attempt to take them on one by one, you're constantly being chased and spammed by the spells of the paladins, so fighting is not quite a viable option.
Starcraft: Terran Mission 3 is supposed to be a hopeless fight, with your goal being to survive half an hour. Unfortunately, the Zerg bases were somewhat less robust than that would suggest. Terran 9, on the other hand, features a genuinely indestructible Zerg base, because destroying any of the buildings ends the mission (It Makes Sense in Context).
In the Starcraft II: Wings of Liberty mission "In Utter Darkness" the ultimate fate of the Protoss (and Zerg) is to be annihilated. Your mission objective is to hold out as long as you can until the inevitable end, where you must be wiped out in order to 'complete' the mission.
Two levels in Homeworld 2 have you facing off against Progenitor Keepers, incredibly advanced drone warships left behind by a Precursor race millions of years ago and who've taken a disliking towards you for having stolen one of their ancient artifacts. The player's most powerful weapons will barely dent a Keeper, and once worn down by attrition they simply hyperspace out and reappear again with full health. It is then up to the player to stay alive against this onslaught until the game script eventually bails him out of the situation.
There is one trick to making them immediately bail - the Scout's EMP blast. If this can be successfully deployed against the Keepers, you can minimize the damage to your forces, as it gives you enough time to burn their health down or they simply jump out on the spot.
The First time you fight Purge the Great in Space Channel 5 Part 2, Ulala is only able to dodge deadly lightning attacks. After giving you some truly Nintendo Hard commands, Purge goes and kills Ulala.
Gitaroo Man contains two examples. On the way to Planet Gitaroo, your ship is ambushed by a giant robot shark, and all you can do is dodge until you unlock a Humongous Mecha to fight back with. Then, the Big Bad kidnaps your dog (who helps you transform) and the player has to dodge attacks from the Sanbone Trio until they accidentally free him for you.
In the very first mission of Chronos Twin DX, while you can be 'hurt' by regular enemies, nothing will kill you until you reach the boss, who exists across two timezones at once. He kills you dead by attacking from the past. The story then shifts forward several years to your character's younger brother trying again, this time with a modified time machine that lets him simultaneously be in the past and the present.
"Versus" at the end of Thunder Force IV's fifth stage. You and your squad shoot it to no avail, and your squad soon gets their asses completely handed to them, forcing your remaining members to resort to give you the Mid-Season Upgrade.
The flash game RPG Shooter: Starwish has the introduction level. You face against some easy cannon fodder Mooks... then you come across the Firebolt. This thing spams bullets and very damaging, hard-to-avoid lasers, and while you can beat it (doing so gets you a different cutscene and an achievement), it's very hard to do so and you'll probably go down.
And if you think you can curbstomp it in a New Game+ with all your cool weapons, it also gets a health and damage buff and is just as difficult!
In the climactic mission, the plot called for you to lose your wingmen in battle with an enemy ace and make the final attack alone: however, this was achieved by having the ace magically respawn so long as any wingmen were present. This lead to a surreal battle in which you might shoot him down a dozen times in a row, using up all of your missiles and countermeasures, and have no way of knowing what obscure action would cause things to proceed.
Similarly, forgetting to use a certain technology could also lead to a constant stream of respawning aces.
Finally, if you fail a critical mission and end up in the losing path, the final mission involves a confrontation with a unique Kilrathi capital ship which is almost impossible to kill without whittling it down with many minutes of firing. The expectation appeared to be for the player character to die trying so that the Bad Ending could roll. As you were not meant to destroy it, the game has no idea what to do when you beat it so just leaves you hanging in space.
While few in the Ace Combat series, "Shattered Skies" in AC04 may be the only one where there are enemy planes that are literally invincible: you'll never actually shoot down a Yellow Squadron member in that mission, although there's a dialogue line if you manage to hit one. There are two missions in Ace Combat 5 where you're actually unarmed (although you're actually very unlikely to get hit), and the series is sprinkled with missions where engaging the enemy aces would run counter to the mission objectives.
Of particular note is "8492" in Ace Combat 5, which pits you against multiple very tough enemy aces flying some of the best air superiority fighters in the game... right after a very taxing close air support mission, not allowing you to change your attacker plane for a more suited fighter jet between the two engagements. So although shooting down all of the enemy planes (as difficult as it may be) does result in mission success, unless you are flying a very advanced multirole jet, any sane person would rather simply flee, just as the story demands.
Technically speaking, Stonehenge in 4 starts out as this. It first causes trouble in mission 7, right after you complete your main objective - it's almost like a Kaizo Trap in this sense, except you receive plenty of warning at first. You can't even attack it at all in this mission, as it's not even anywhere on the level and is instead firing One-Hit Kill shots from afar. You have to enter a tight ravine to keep out of it's sights, and exiting the ravine (a.k.a. going above 2000 feet) is practically suicide. At that, you only know shots are coming ten seconds before they hit. Later, thankfully, you're given a warning to expect it to act up, and you get more time between shots. You still can't even fight Stonehenge, however. Fast forward to mission 12, and you're actually able to attack it and know where the cannons are, making it far easier to manage. Watch where you're flying though, because the cannons still deal out One-Hit Kills with impunity, and you don't wanna be right in front of one when it goes off.
In Star Trek: Bridge Commander, there are a few times where you are encouraged to complete your objective and get the hell out of there ASAP, due to being highly out-gunned. A few times, ships respawn infinitely, but a few times you're highly rewarded for defeating all the enemy vessels.
In Descent: Freespace you can fight the Lucifer multiple times, but can only destroy it during the last mission due to its invulnerable shield. (This is because the final mission is the only time you encounter the Lucifer in the campgain where the "Invulnerable" flag for the Lucifer is not turned on. Yeah, that's right, the Lucy doesn't have shields, just a mission editor hax.)
Not that you'd stand much chance of destroying it even without the shields: you're a small, one-man fighter and the Lucifer is a 3 kilometer long superdestroyer. It takes a full squadron of fighters guarding two wings of heavy bombers armed with bombs meant for planetary bombardment to destroy it in the end.
On the World of Tanks Russian server, fifteen T-34-85 drivers can enter a boss battle with the White Tiger, driven only by WarGaming staff. It has 14,960 hitpoints, making it Nigh Invulnerable, and uses the L/56 8.8cm but fires 15cm shells from the E-100. Released partially to promote the obscure German film "The White Tiger".
Stealth Based Game
In Metal Gear Solid 2 Raiden goes up against 'Lady Luck' Fortune. Any shot fired at her misses, and any grenade thrown near her is a dud. Your only hope is to dodge her shots until events elsewhere force her to leave. The battle with the Metal Gear RA Ys counts as well, since Raiden eventually tires from the fight.
In Metal Gear Solid 3, when you encounter "The Sorrow" any sort of offense is useless since you can't hurt him or his spirits. The only way out is by convincing The Sorrow that you died, game over screen and all.
The DLC of Assassin's Creed III has one against George Washington. Yes, THAT George Washington. If you run up to him, he blasts you away with an energy wave, if you run circles around him he'll shoot lightning at you, if you just keep your distance he'll summon thunder bolts from the sky on top of you.
Silent Hill 2 and Silent Hill 4 both feature unbeatable bosses. Silent Hill 2 has the ever-popular Pyramid Head, and after running away from him down a long, narrow, winding hall, blindly shooting at him, you run into an elevator to escape. Not everyone does though, and he manages to kill Maria. But only sort of. Eh. In Silent Hill 4, Walter Sullivan chases you for the entire second half of the game. He cannot be killed, but he can be slowed. He and PH do, however, eventually become beatable.
The two actual "fights" against Pyramid Head (in the apartment building and near the end of the game) definitely qualify. The only way to end these fights is to either die or wait until he decides to end it after a certain amount of time ( the first time by leaving the room, the second time by both Pyramid Heads impaling themselves on their own spears). However, while you can't actually "win" the fights, shooting him does shorten the amount of time it takes Pyramid Head to end the fight.
Inverted in Silent Hill: Downpour. At the end, you get transformed into the Bogeyman (since this is how Anne has viewed you during her journey through the town.) You are essentially a hopeless boss fight for her, and can choose to smash her brains in with your huge hammer or spare her. Her pistol does Scratch Damage to you, so you can stand there and let her kill you for a Twist Ending.
In the Gamecube Remake of the original Resident Evil, Lisa Trevor is an invincible deformed implacable girl super-zombie that stalks the player in several areas of the game. The final encounter with her is a Puzzle Boss fight where you have to push several blocks off a ledge to open her mother's coffin, causing her to jump to her death at the sight.
Lisa Trevor is a subversion. She seems like a hopeless boss, but if you have ammo to spare you can knock her off the edge by shooting her enough.
Even then in the Wii rail shooter Umbrella Chronicles Lisa apparently survives her fall/suicide and becomes an unbeatable boss against Wesker during the Rebirth scenario. The least he could do against her was slow her down and pin her under a chandelier. It takes the mansion's self destruct system to finally kill her off.
You can turn around and shoot him to slow him down (the explosive arrows are best for this). Technically, you can get away without a scratch. However, it is not possible to defeat him.
The first time you encounter a Leech Zombie in Resident Evil 0, you lack the firepower to defeat it, so you can only run for the exit, which triggers the cutscene where Billy takes it down.
The T. Rex in the Dino Crisis series is an unbeatable boss, and if he catches up with Regina, he will swallow her whole. Although he cannot be killed, he can be slowed down with gunfire.
The first few encounters with the White Witch from The Dark Meadow will be this. You simply don't have the proper equipment or experience levels to deal with her, forcing you to investigate more of the game's story and kill more monsters first. And don't even think about winning against The Trickster's second and third forms on your first try.
The zombified Alan Morton, the final boss of Alone In The Dark The New Nightmare. No matter how many times you knock him down, he always gets back up. After stunning him, you have to run into a mundane alcove which contains a spear, which Carnby automatically uses to kill Morton.
Likewise, Obed Morton is so difficult to defeat that to many players he seems like a Hopeless Boss Fight, such that even most of the game's walkthroughs indicate the only way to finish the game is to use an exploit to run past him instead of fighting him. You actually can kill Obed by shooting him (and it doesn't even matter what weapon you use either), but you can only hurt him when he's in a certain pose (he should be knocked backwards if you hit him correctly, indicating you got it right).
In Fatal Frame 2, you have the Kusabi. An extremely powerful ghost, any time he appears before the end-game fight against him, he's immune to the effects of the Camera Obscura (rendering him invincible) and getting touched by him is instant death, whether you have a Stone Mirror or not. Your only option is to run.
The same game also features a sequence where the player must run from Sae Kurosawa, another extremely powerful one-hit kill ghost. In this case, though, she's invincible because the player has lost the Camera, and with it their only means of defending themselves against ghosts. Curiously, when she's encountered as the True Final Boss, she's no longer able to one-hit kill you.
The Fatal Frame series makes a tradition out of this: Kirie Himuro from the first game is also invincible and a one-hit kill until the final boss fight. The third game has Reika Kuze, who does not have one-hit kill powers, but is invincible until (you guessed it) the final boss fight.
Turn Based Strategy
Final Fantasy Tactics Advance has two of these, both of them involving Llednar. He's invincible and strong. You have to stall for several turns before the plot takes over. He does become beatable towards the game's end. Just to give an idea of how unbeatable Llednar is: Most battles have objectives like "Win battle!", "Defeat the Boss!", or the suchlike. During the fight that's just Marche versus Llednar? "Survive!".
Final Fantasy VII: Crisis Core also has one of these, during the final battle between Zack and SOLDIER as he fights to obtain his freedom and protect Cloud, it is literally an impossible fight to win, the player is required to let Zack die at the end of the game. Then, just to add insult to injury, if the player manages to stay alive in the fight long enough, the game will actually begin firing missiles from the tanks in the background of the battlefield that are nearly unavoidable, and if they hit Zack, it immediately drains him to 1HP, triggering the final cutscenes and ending the battle.
Disgaea 2 subverts this at one point. Initially, Laharl shows up and kicks your ass. However, in the process he breaks Rozalin's talisman, triggering her Superpowered Evil Side. This causes an inversion, where it is he who cannot possibly defeat you, however, you HAVE to lose, if you win this battle at the first phase then you get an ending and are forced to start the story from the top. The same with your battle with Etna. If you lose, the game continues, if you win, you get a joke cutscene, the credits roll, and you have to take it from the top.
Another Disgaea subversion happens during the first game. Initially, during the Chapter 6 fight against the Alternate Overlord, his group of ten is all at level 75, and you're normally barely breaking 30 on your first playthroughnote Meaning that should you lose, the castle retainers, their levels around 300 to 400, show up to bail you out as Neutral NPCs. However, should you win this fight on another playthrough (or just level grind enough on your first time through), it turns out that the game just proceeds on. No special cutscene, nothing.
The Nippon Ichi game La Pucelle Tactics also had two such bosses, in chapters 4 and 8; defeating them earns a "special ending" for the chapter which grants a powerful piece of equipment.
Storywise, the only World Eater that falls under this category is Feinne, the first World Eater you encounter (right at the beginning of the story, to boot). Actually beating her triggers the battle against Asagi.
Rhapsody: A Musical Adventure loves to use this trope. Several boss fights (primarily with Marjoly's minions in early goings of the game) are impossible to win, simply because you can't damage them - all attacks will miss). Others later you have to at least survive for a few rounds before the game ends the fight. Still, the most amusing one is during the contest when you fight Etoile. You vs Rocket Launchers and Machine guns. Good luck!
Makai Kingdom actually inverts this at the beginning, placing you against a series of opponents whom you could not possibly lose to.
Chapter 8 ends with Alexander finally stepping to defeat you on his own, being way above your level on a first playthrough. Fortunately Salome joins the fight as an uncontrollable third party, even higher level, and turns him into a smear. Unless you have the bad fortune to get into the crossfire, it's a sure win. And of course, on future playthroughs you can fight them on your own for certain alternate endings.
In Super Robot Wars: Original Generation 2, you are constantly pitted against the main bosses of the game — who retain their endboss-level stats the entire time. Whenever they show up, you're supposed to simply stay away from them until they leave, but one battle forces you to go all-out on one who is strictly unbeatable. In another, similar situation, the boss won't leave; either you leave, or you finally beat it by what amounts to cheating.
Should you actually win the first fight against the Inspectors (the Quirky Miniboss Trio), you get an awful lot of very good items, ridiculous amounts of gold and exp, and a rather amusing No Fourth Wall moment.
In Original Generation 1, there is the early and optional fight against Shu Shirakawa's horrendously broken Granzon. You are encouraged to run, although battling it is an option and a viable one at that, provided you have put a lot of money and level-grinding into Irmgult and the Grungust. Even then, the Granzon still takes a beating and only your top-tier units at the time should be used against it.
Warsong (Langrisser in Japan) had a non-boss version of this. In the first scenario you are supposed to escape, but a clever strategist can beat all the enemies instead. This has no effect on the plot except for a sentence or two.
In Fire Emblem, Captain Fargus challenges your group to see if they're worthy of being given passage to the Dread Island. If you select "Attack" instead of "Talk" when you do reach him, he will fight the character you used to challenge him. Since he's a promoted class as well as extremely powerful yeah, you can guess the outcome.
Valter in the beginning of Sacred Stones is completely unbeatable, as was demonstrated on youtube to comediceffect.
The Invincible Black Knight in Path of Radiance shows up in an earlier mission when you are expected to not even have a promoted unit yet. Your goal of the mission is to escape the town by sea and if any, and I do mean ANY, character attempts to hold the Black Knight off...Well, they had a good run.
Radiant Dawn does something similar where the Not Quite Dead Black Knight is on the opposing side of the field in Part 3. Fortunately, he doesn't move until about 2 turns before the battle ends, which means he's no threat unless you deliberately step into his attack range. However, it does play with this Trope by offering you an interesting reward for challenging him; if Ike has a battle with him and SURVIVES (which requires a very high level at that part of the game), Chancellor Sephiran is playable for the Final Boss.
To make things more interesting, if you equip a strong enough unit with the Hammer-a weapon strong against armored units-it's actually possible to KILL the Knight(He's coded with the hammer weakness, even though he's only supposed to be vulnerable to certain blessed swords and magic). Doing this disqualifies you for the reward though, and he's not really dead anyways.
Battle Moon Wars, Stage 27: when it gets down to Weak!Shirou versus Archer, the game expects Shirou to hit 0 Hit Points and for the Shikis to have their Big Damn Heroes moment. However, it's possible without cheating to beat Archer; well, it would be, except when he hits 30% of his health, the script goes on as if Shirou had been defeated. You barely get a line of acknowledgment for your troubles.
Devil Survivor, Day 3: Beldr, first encounter. When NPCs say that Beldr is immortal, they mean it; his HP and MP are "???", and he nullifies everything,Almighty included for wiseguys on their second playthrough. Your only option is to escort Midori out of battle, then have your party escape.
Devil Survivor 2, Day 7: Benetnasch, first encounter. It possesses a passive skill called Pacify Human, which naturally cancels out any attack from a human character... and then it pulls out the ability to forcibly unsummon your demons, leaving you no way to harm it and thus no choice but to run away until you can figure out a way to counter it.
You also can't harm Dubhe at first - the objective is to reach the other end of the map and escape, and losing any party member in the process results in an ordinary game over. However, just as you reach the escape area, it becomes blocked off and a cutscene results in Dubhe becoming vulnerable.
Sengoku Rance: If you fail to stop Miki from becoming the Demon King, you will enter the Demon King pseudo-route, and a Hanny will appear to your castle and tell Rance that there's absolutely no way to win. The enemy army will become incredibly tough by enabling all 4 buffs for every unit, greatly increasing troops sizes and fielding Kentarou and Xavier units with 8000 troops each against you. It's still possible to actually conquer all the enemy territories by using battle permits, but it will lead to Kentarou accusing the player of cheating and a standard Game Over anyway.
The first Sakura Taisen game had this in the first battle with the Koumas, where you're expected to fight for about one turn, realize your attacks are doing almost nothing, and run. In the PS2 remake, they're actually beatable, but the cutscene still treats it as if you ran away even if you stay and beat them.
In the Descent to Darkness campaign in Battle for Wesnoth, the final mission is essentially a hopeless series of boss fights. Mal Keshar is attacked by a hero and his/her army - it isn't hard to defeat them, but the mission will just repeat itself with new heroes until Mal Keshar is slain.
Ge Ne Sis: Reciful. The guy has 80,000 HP and reflects half of the damage dealt to him back to the attacker. However, you do get to fight him again straight afterwards using Esis, who is more than capable of dealing 80,000 damage in a single move, which is extremely satisfying.
Turn Based Tactics
At the end of Transformers G1 Awakening, the player is made to think that Shockwave must be defeated by Optimus Prime and Bumblebee. However, both Autobots are significantly weaker, even working together, and there are no power plants on this level that would allow them to level up. However, after Prime goes down, The Cavalry shows up in the form of Grimlock, who proceeds to Curb-Stomp Battle Shockwave.
In Bahamut Lagoon you have to fight against Sauzer/Zauzer in Chapter 5, but don't waste any items or energy, this guy has infinite HP, so you can't kill him.
Wide Open Sandbox
Shenmue has a character named Chai who is supposed to beat you; when you lose you're unconscious for several days, but the game isn't over. Beating him is nearly impossible, but can be done.
Shenmue 2 has one hopeless boss fight where Ryo fights Dou Niu and his henchmen in a plaza until the screen blacks out and shows a cutscene of Dou Niu beating Ryo.
Grand Theft Auto2 featured a mission that required you to "die" while attempting to break into an enemy gang's stronghold. Your employer then picks you up at the hospital and sends you back to the fort with a better plan and better equipment.
The original Grand Theft Auto Classic featured a mission where you're asked to board a train as part of a plot to prove that your boss is the one true god. It's then revealed that the train is rigged with explosives. If you stay on it, after you die your boss apologises, saying that he was high on smack and is impressed you made it... Assuming that wasn't your last life. Worth getting off the train and failing just for the quote: "Only the righteous shall be saved! You! You are a shithead and bound for hell!"
In inFAMOUS 2, the very first battle in the game is against The Beast, a being of overwhelming power that Cole is trying to become more powerful to defeat. In a surprising inversion, the battle is far from hopeless, as Cole ends up winning. Then the Beast breaks Cole, literally, and robs him of most of his advanced powers before breaking apart. It then reforms in Empire City, destroying it after Cole is taken south by boat, and the real game begins. Turns out the Beast can regenerate himself after every defeat, including after being struck by a nuclear weapon halfway into the game. The eventual method for defeating him is to kill him on an atomic level using the RFI.
The sequel to the above, inFAMOUS: Second Son, has a story-related twist on this for the DUP King boss. The idea is that it's a Hopeless Boss Fight so long as the hero, Delsin, sticks to using Smoke powers and stubbornly refuses to use his Neon ones. It is expected that the player will eventually run out of Smoke energy and be forced to use Neon, however skilled players on lower difficulties can in fact beat the DUP King without having to do that. This causes dissonance in the next cutscene as the characters will behave as if Delsin gave in and used Neon.
In Red Dead Redemption has the final "boss" of the mission where you kill waves of army and BOI agents ending with John making a final stand against a line of soldiers as his family flees, getting shot to death by the same man he helped track down all of his old friends. Oh, the irony.
Non-video game examples:
Anime and Manga
This trope is parodied in the Suzumiya Haruhi side story "Haruhi Theatre Act 1". Haruhi and Brigade are trapped in an RPG, and when Haruhi meets a sage who warns her that she can't defeat the dragon without the power to do so, she doesn't listen and drags the Brigade over to fight it. Naturally she gets creamed and regenerates next to the sage, who smugly tells her off. She doesn't listen again and continually goes back and dies. When she finally listens to him, he demands some stuff from her as he's only giving them out if she gets quest items... but she just threatens him.
Homura's entire battle with Walpurgis Night in Puella Magi Madoka Magica ultimately turns out to be this despite the truly epic level of firepower that she brings to bear, and in her incredibly heartbreaking Despair Speech, it's made quite clear that this isn't the only time she's failed to defeat it on her own.
Discussed in detail in the Star Trek novel Kobyashi Maru in which we see exactly how Scotty, Chekov, Sulu, and Kirk all dealt with the eponymous training simulation: Scotty performs engineering miracles until the ship is finally overpowered and destroyed (this leads to him being transferred out of Command School into the Engineering Corps), Chekov rams the Enterprise into the Klingon ships, destroying them (and the Maru as well), Sulu realized that it was likely a trap and left without crossing into the Neutral Zone, while Kirk reprogrammed the simulation so that, when he introduced himself, he was so well-known as a badass that the Klingons immediately surrendered.
In one famous week-long series of FoxTrot strips, involving a Fictional Video Game, Jason has spent a month playing a game trying to beat a Boss called the Red Orb Guardian who he can't even come close to defeating. When Paige tries to play the game, she figures out why; you aren't supposed to fight the guy at all, if you simply walk past him, he won't stop you. (Jason seems to wonder why you aren't supposed to challenge a huge, hulking, ferocious-looking warrior that has skeletal corpses surrounding him, calling the situation "counter-intuitive". Of course, Jason is like that a lot.)
The whole point of the Kobayashi Maru test in Star Trek. Like the Giant's Drink above, it's actually a psychological test rather than a tactical puzzle, although that doesn't stop people from trying. As various cadets' attempts to hack the infamous sim have piled up, they've resorted to ever-more-obvious tactics to keep it hopeless, including spawning extra enemies when needed or allowing them to break the laws of physics.
In one Vampire: The Masquerade rulebook, there is a section on fighting Caine that consists of two words, "You Lose." He is the Biblical Cain, turned into the first vampire by God. He knows every vampiric Discipline in the books, has centuries upon centuries of experience, and has the ability to do - more or less - anything he wants. In addition, he possesses the Mark of Cain from the Bible. Any damage inflicted on him is returned seven fold to his attacker, and on the off chance someone did manage to kill Caine, they would instantly die.
Togashi Yokuni in Legend of the Five Rings. The utterly mysterious lord of the Dragon Clan, who is so enigmatic that those who speak with him don't even really remember what his voice sounded like or what words were said, is actually Togashi-Kami, the immortal child of the Sun and Moon and the only remaining sibling of the first Emperor. He is also a dragon. He can see the future and knows when he will die, and it sure ain't gonna be you that kills him.
The Lady of Pain in Dungeons & Dragons. She is the enigmatic steward and protector of Sigil, the City of Doors, whose motives cannot be divined and who can (and has) annihilate even higher tier deities with her power. She will flay you outright or trap you into a nigh-inescapable labyrinthine dimension should you ever do anything to cross her. Stats for her simply aren't given, because She > You no matter how many epic levels you've got. And if you're a minotaur with an adept sense of direction? She just puts you in a single hallway. One. Long. Hallway.
There's also the entity known as the Guardian of the Dead Gods, a being who was formerly the god Anubis, who now watches over the dead gods in the Astral Plane and protects them from desecration. It's very hard to tell just what will make him intervene (the githyanki have built entire cities on the giant corpses and he doesn't seem to mind, but sometimes, simply touching one might get him upset). The article where he's mentioned gives a flat 5% chance that doing anything to one of the dead gods will attract his attention (unless someone does something really stupid, like try to completely demolish one, in which case it's almost assured), and even then, he'll usually send his servant to deal with it (who's a powerful wizard, but not unbeatable). If you're unlucky enough that he decides to deal with it personally... The same source says that your only chance is to flee (he usually will not follow, but there's no guarantee). Simply put, he cannot be hurt by anything mortals have at their disposal.
This was a common theme in Planescape in general. While on a Prime (Earth-like) world high-level PCs might get used to being the biggest badasses around, on the planes there's always a bigger fish - always. Demon lords and archdevils were the weakest of the untouchables (who you had a chance of at least defying and defeating their servitors), then any of the gods were next up the list (with their own hierarchy from hero-deity and demigod to greater power), and finally up to enigmatic cosmic forces like the Lady (who can kill gods) and the Guardian of the Dead Gods. At the same time, Planescape was dedicated to cutting a survivable niche out for the merely mortal while in realms populated by such beings, when before the planes themselves had only really been open to high-level characters with powerful magic.
In one of the Warrior Cats tabletop games included with a few of the books, if you attempt to fight a Twoleg, it ends like this. The PCs' only options are to attack, which does nothing but damage them, or run away.
In Sla Industries, two unstoppable badasses are repeatedly made mention of: the infamous serial killer, Halloween Jack, and the king and father of all Manchines, Digger. While both are a notable threats or inconveniences to the company, and have many attemps made against them, both are effectively unkillable: Jack has has a bounty of 2 billion credits on his head that no sensible op would try and collect on, and Digger has a whole army to fight with. Niether character has stats in any of the books, as they are considered to OHKthe party if they ever cross paths. This especially bad news for those taking one premade platinum mission, which requires them to go right into Digger's homebase, and sabotage his ultimate plan to conquer Mort.
In Arkham Horror, failing to seal enough portals before too many have opened will result in the Big Bad of the session to break through and the players facing it in direct combat. The fight is heavily stacked in its favor and the PCs are probably going to die, but it's potentially winnable... except when the Big Bad is Azathoth. If Azathoth wakes up, the whole universe ends.
In Warhammer 40,000, if your troops are stuck in close combat with a Monstrous Creature or Walking Tank, and none of your weapons can damage said Creature or Tank, they may choose to run away automatically by declaring "Our Weapons Are Useless!". Just keep in mind the Creature or Walker gets a chance to kill them instantly, but a half chance or so of escape is better than nothing.
Swashbucklers of the 7 Skies ranks NPCs from cannon fodder Minions all the way up to powerful Archvillains, but then notes that some characters are so arbitrarily powerful they can only be designated "Plot Devices" and so doesn't bother giving them stats - for example, Alexei Rostand, a retired sky-pirate who had been at the work around 50 years before he managed to win a pardon. Even Rostand's lieutenants go un-statted and are simply described as extremely dangerous.
Erf World is this for Parson, where he gets sucked into a game similar to one he designed, and realizes that since he wouldn't let his players win without cheating somehow, he won't win his battle either. At least not without trickery. He uses every loophole and unorthodox tactic he can think of to eventually defeat the opposing force.
Parson: "That, um, game I was developing at home... it had rules. But it couldn't be won within the rules. I wanted a game where the players had to surprise the GM with lateral thinking. So I was essentially going to cheat them. Undermine everything they tried. Until they found a clever enough way to cheat me. To break my rules, and win."
Parodied in College Saga. The characters face Leviathan, who uses the devastating "Tidal Wave" attack, and quickly kills off the party... and then says "Just kidding LOL. You can't kill a guardian force."
The Aztecs had a ritual for Huehuetéotl in which a slave was tied almost naked to a rock and given a blunt weapon to free himself and fight against three heavily armed warriors.