"The opportunity to secure ourselves against defeat lies in our own hands, but the opportunity of defeating the enemy is provided by the enemy himself."
— Sun Tzu
In video games, it's quite common for a boss to only be vulnerable at a particular time. It's one thing if it's due to the boss needing to catch their breath every so often or the like. But sometimes, it's just because a given attack leaves them open. Maybe it exposes a weak spot while preparing the giant laser, maybe their attacks double as weapons or platforms that the player can use to reach their target, maybe on occasion it decides to just stand there laughing at you with its defenses down, or maybe it will keep charging head on into wallsuntil it dies.
This is a specific form of Hoist by His Own Petard - the boss only has to avoid one particular action to be literally invincible. Since it involves the boss intentionally leaving themselves open, it also heavily involves carrying the Villain Ball. Finally, Fridge Logic heavily comes into play - it's usually in retrospect that you realize What an Idiot the boss in question is for continuing to perform the attack even after it's been exploited several times. With animal bosses or others too unintelligent to realize their mistake, this can be justified, and to an extent it can make sense with robotic bosses as well, but in that case you have to wonder who designed and programmed them...
Hopefully, sooner or later there will be a variant of this where the boss does stop using a particular move that leaves it open to attack... only to then use a different move that also leaves it open to attack.
Bosses like this can occur with any of the three types of AI (as described in Artificial Stupidity)
A.I. Roulette: The boss will use one of several attacks randomly, but only a fraction of them (often just one) leaves it vulnerable to damage. Since the length you have to survive for is dependent on how often the boss makes the suicidal attacks, the fight is partially a Luck-Based Mission.
Set Pattern: The boss has a fixed cycle of attacks, and at some point renders itself vulnerable.
Analytical/Responsive: The boss will make itself vulnerable if given a certain situation. If the boss was run by a smart analytical AI, it would only use the vulnerable attack as a last resort, when no safe attack is possible (possibly not even then, if the boss doesn't care about letting the player live). But usually Artificial Stupidity is applied deliberately to prevent the game from becoming Unwinnable, especially if the boss always has safe attack options available. Alternatively, there may be a Tactical Rock-Paper-Scissors system where the boss defending itself from one kind of attack leaves it vulnerable to another, forcing the player to work out the right Combo of attacks to defeat it.
When you confront GLaDOS again, she's taken care to keep you from using the same tactics to beat her as in the first game. However, unbeknownst to her, you've already sabotaged her defenses, and the mainframe helpfully provides a third option to the ensuing stalemate.
At the start of the final battle, the boss informs you that it has analyzed the first game's climax and set up an arena with no portal surfaces and thus no way to fight it while the neurotoxin kills you. However, it took the extra step of equipping itself with bombs (and reconfigurable shields so you can't hit it with the bombs), but forgot about the tube of Conversion Gel running through the room. One broken tube later and you gain the means with which to defeat it. What's more, after you hit it with a bomb for the first time, it tries to stop using them but discovers that the mechanism for doing so has been damaged.
Super Mario Bros: Bowser mostly commits Boss Arena Idiocy, placing an axe at the end of his bridge even after seven fakes fell to their doom. But for this trope, his habit of making big slow jumps in the air can be incredibly helpful to the player, especially in later worlds.
Super Mario Bros 3: Bowser will keep ground pounding at Mario, even if it means his death.
Super Mario World: If Bowser stopped throwing Mechakoopas after Mario, there'd be no way to defeat him.
In New Super Mario Bros Wii, if Bowser just restricted himself to smashing the foundations of his castle and not fire breathing, blowing out passages, he would have killed you.
Hell, this goes for almost every boss in the game. If they would keep spinning, Mario and company would eventually tire themselves out. The only exception is Iggy and Ludwig, and only the second time you encounter them (Iggy cause he just gets stunned and Ludwig cause he's stuck on a small platform while doing so), as well as bowser jr., who has different reasons for his screwups.
Except in the DS "prequel", where it's played straight in about half of his fights. In most of the later fights, he becomes aware of the fact that he's vulnerable when not inside his shell, so he retreats into it whenever it seems like Mario might jump on him. (You can tell when he's doing this when his bandanna is over his mouth) Unfortunately, he throws Koopa shells at you after every time he does this, which you can throw back at him to flip him over helplessly. Not the best compensation for a new tactic.
In Super Mario Galaxy, if Bowser just restricted himself to breathing fire and spinning around in his shell, you'd have no way to defeat him. Thankfully for the player, he'll also jump at you, even if you're standing on a thin layer of glass which is right above a lava pool... which is the only thing that can damage him.
Averted by, of all things, Baby Bowser in Yoshis Island. About the worst thing he does is get too close to Yoshi, making it easier for himself to be hit; other than that he doesn't do anything that contributes to his defeat.
The Post Final Boss of Mario And Luigi Partners In Time is a very special example. Shrowser fires some fireballs and counters others in an absolutely horrible way. He ends up attacking his new power source: the ghost of the Elder Princess Shroob.
Pretty much all Super Mario WorldGame Mod boss battles are one of the first two, with the ones in games like Brutal Mario, Scarlet Devil Mario and Super Mario LD being set pattern types and the ones in certain other hacks being the AI roulette types.
In Super Meat Boy, many bosses do this. For example, the boss of Chapter 1: The Forest, Lil' Slugger, will chase you through a sea of sawblades, and ultimately commit suicide in a sawblade trap. Little Horn in Chapter 4: Hell, will smash his head into the small platform you stand on. If you dodge it, Little Horn gets a bump on his head and is damaged. Two more times and he's dead. And, as a reference to Super Mario Bros, in the final boss fight with Dr. Fetus, you get to press a button and watch the floor collapse under him as he falls to his death(?).
In Gradius II, the Crystal Core could render itself beatable by just wrapping its tentacles around its weak point. But it doesn't.
In Gradius Gaiden, Kraken, the Stage 7 boss, could have been able to make itself near-invulnerable if it kept its arms closed and never used its giant rock attack (which exposes the core barrers on its rear).
In No More Heroes and to a certain extent No More Heroes 2: Desperate Struggle, if you think about it. All of the bosses in the first game are actually completely invincible or extremely evasive most of the time. There are usually only small windows of opportunity where the boss is actually vulnerable to your attacks; learning when these windows appear and exploiting them is pretty much the only way to win the game. In the second game, you can attack whenever for most bosses, but attacking at the wrong time can cause you to get hit.
As Dethl's Agahnim form is a deliberate recreation of said fight, this also holds true in Link's Awakening.
The Agahnim clone in Oracle of Seasons can be killed this way, but since he's vulnerable to the sword that time it's not the most efficient method.
A LOT of bosses that do this: King Dodongo (keep rolling, don't spit fire), Ganondorf (almost every single fight, he shoots an energy ball that you two end up smacking back and forth; spam your un-reflectable powers), Molgera (dude.... seriously? Sticking your tongue out and just sitting there?), Morpha (just stay under the water, don't make yourself an obvious Clawshot target), Koume and Kotake (stay on the other side of Link when he's reflecting the ice/fire magic, don't fly in front of it; alternatively, aim at the ground beside him for Splash Damage Abuse) and others that I'm blanking on right now.
In Oracle of Seasons, Onox, the final boss is this in his third phase: he is a flying dragon, and his sole weak spot is his forehead that is too far away for Link to hit from the ground with his sword. Similar to Sigma from Mega Man X, he makes the forehead vulnerable by slamming his claws into the ground, when he could spam his hard to dodge flame attack.
The many-armed swordslinging robot Koloktos in The Legend of Zelda: Skyward Sword can only be damaged by his own swords. He'd be unbeatable if he didn't keep swinging his swords into the floor so violently that he gets stuck, allowing you to rip his arms off, take one of his swords, and beat him up with it.
King Dedede has these attacks which makes stars appear. If he never used them then Kirby wouldn't be able to spit the stars at him. (But in the later Kirby games, Kirby can simply bring along a special power and pummel him.)
Virtually all bosses and sub-bosses in the Kirby games have at least one attack that causes an inhalable object to appear, most often a star. The only exception is Meta Knight in Kirby's Adventure, since the player was given a permanent Sword ability for that battle. Meta Knight himself gave you the sword, but he hashis reasons for that.
Kirby Super Star has the Reactor boss from "Revenge of Meta Knight", which is invulnerable to everything except its own lasers reflected back at it.
There is one boss from Just Cause that decides to attack you with a Kill Sat that shoots missiles, while standing on the same narrow rooftop as you. To quote from Zero Punctuation, "Didn't even have to shoot the stupid bastard. Would have practically been assisted suicide."
Many of Robotnik/Eggman's machines are like this, especially in the later games, where he has attacks that can hurt Sonic without going into harm's way, but he does so anyway. He has an IQ of 300.
The final boss of Sonic 2 is a huge mecha with a small hit area, and a claw that completely covers it. If Robotnik just walked, he would be invincible (or at least extremely hard - he has a miniscule hit zone during his walk animation and everything else kills Sonic). Instead, he jumps, making his claw sway to regain his balance.
The Flying Battery miniboss can only attack you with its oversized spiked flails. It doesn't try to think up a better way to deal with blue hedgehogs on its head other than bashing its own head in. Even assuming that it's not mechanically capable of low lateral swipes over the top of its head, it would still be able to win by doing nothing, thus keeping Sonic busy long enough for Robotnik to finish carrying out his evil plan.
In the Lava Reef Zone boss fight, Eggman will continue to shoot spike balls while inclining the surface such that they roll into his own ship.
The first phase of the True Final Boss can only be undone by its own heat-seeking missiles.
Knuckles's final boss, Mecha Sonic, charges itself up with the Master Emerald. It's invincible, but the charge eventually fades away, making it temporarily vulnerable while it retreats to the emerald to recharge. Later on in the fight, it starts to look heavily damaged, and can't maintain the emerald's charge, forcing it low to the ground while hovering right in time with the invincibility fading away for a moment...
The Egg Viper. During the first part of the fight, Eggman regularly uses a charged laser attack that requires him to hold still with the cockpit wide open for several seconds; most players will never even see him get to the actual attack. He later switches to a slightly more direct attack that involves shooting a very platform-like projectile that Sonic can easily ride straight back for the hit.
Also, E-101 MKII. He simply bats away all your shots or strafes out of the way, but completely exposes himself every time he attempts to impale you. If he never charged, he'd be invincible.
The Egg Hornet. If all Eggman ever did was shoot those little missiles at you, and never stuck himself in the ground trying to hit you with his ship, Sonic would wear out and eventually get hit.
Chaos isn't immune: in his 2 form, he bounces around as a giant ball in one attack, and covers most of the floor for a couple of seconds in another. Why would he ever need to do that stretch-punch thing? Well... because if he didn't, he couldn't stop moving afterwards, giving Knuckles a chance to punch his brain.
And his 4 form is worse; the only time he's vulnerable isn't even associated with an attack. If he'd stop just surfacing and sitting there, Sonic, Tails, or Knuckles would eventually lose.
Chaos 6 has this problem as well, though to be fair it's not Chaos's fault this time but rather Eggman's, if he wasn't flying around dropping bombs then Chaos would have been impossible to beat, if only Eggman had the sense to stay out of the fight.
In Knuckles Chaotix, what stage the player goes to next is determined by a roulette-like device with an image of each of the five stages on it. As each stage is complete, the images are replaced with 1, 2, 3, X, and 4; once all five stages are completed, the roulette is used to fight Metal Sonic. Whatever number the roulette lands on during the fight determines the attack, but landing on the X causes damage to Metal Sonic. To Metal Sonic's credit, when there's only one number and the X left, X causes the attack and the number defeats him, in case the player is able to time the roulette at that point.
It has bosses like this that use A.I. Roulette. The programmers were considerate enough to actually adjust the time bonus based on when the boss made itself vulnerable. If it did another move, the time it used up didn't hurt your time bonus.
While the Ghost Titan is already one of the most frustrating bosses in all of Sonic history, this Sonic Rush Adventure boss would be literally invincible if he'd just stop shooting and only punched.
King Hippo. If he never opened his mouth, he'd literally be unbeatable.
In the Wii version, interestingly, this wouldn't save him — hooking his stomach when he jabs gets you Stars, and Star Punches will hurt him mouth open or closed.
Everybody would have a better chance against Little Mac if they didn't have tells, stop to taunt, etc. Of course, they do recognize this in time for Title Defense mode, shoring up some of their vulnerable spots (and sometimes opening up whole new cans of this).
He is only vulnerable after you get his various heads drunk. You do this by drawing the sake below him into his mouth when he roars, and he roars after you thwart his attacks. You'd think he'd figure out to keep his mouth closed. Furthermore, you can only truly damage him after knocking out all eight of his heads, which only stay down for a short time. Nearly all the other bosses in Okami fit this trope as well. At least mythology establishes that Orochi is an alcoholic, justifying his inability to stop drinking sake.
It gets better though, because Orochi manages to turn himself into a case of Too Dumb to Live: You fight him three times, and each time he uses the 'Attack-'n-Roar' technique. You would think that after fighting Amaterasu in the past and present, he would have remembered how he'd been defeated, but no. He does the same thing on the Ark of Yamato!
Also, Ninetails. In the first part of the battle, drawing lightning to her sword when she lifts it up for a charged attack is the only way to make her vulnerable.
In American Mc Gees Alice, the Voracious Centipede is invulnerable to attacks, except for an old scar on its underbelly. It is usually hidden but can be targeted when the Centipede is rearing for a body slam attack.
King K. Rool is invincible... until he throws his crown at the Kongs, which allows them to jump on his head and hurt him. He repeats that move 10 times in the battle.
K. Rool in Donkey Kong Country 2 shoots spiked cannonballs at the player, but will sometimes shoot out an ordinary cannon-ball which can be thrown back at him to jam his gun. Nine times. (Ten if you count the True Final Boss). He's even dumber in the third game.
In 3, the battle against him overlaps with Boss Arena Idiocy; you can only damage him with barrels, which are dropped into the arena from levers which become available as he attacks. Kaos and Barbos from the same game are straighter examples; Kaos' blade spin attack does nothing except provide a handy platform from which to jump on him, and if Barbos didn't send his minions after you, or launch homing missiles, you'd have no way to damage his shield.
Donkey Kong 64 makes it even worse. For example, when fighting him with Tiny Kong, he will repeatedly butt-slam the arena to cause shockwaves. Eventually, his butt gets too sore to keep doing it and he moans in pain for a while, leaving him wide open to attack.
Some bosses in Fraxy will flip open to expose their weakpoints after some time, then close them again. Some attacks are also tuned to start when this happens, so watch out!
If Kraid would keep his mouth shut he would be invulnerable. Justified in every appearance after the original game in that he must be deliberately provoked into doing this by shooting him in the eyes, making him roar in pain.
If Metroid Prime would not squirt out pools of pure Phazon, Samus wouldn’t be able to use the only weapon that works against it.
If Phantoon would stay phased out and shooting at Samus instead of periodically materializing in the middle of the arena, he’d be untouchable. Though Phantoon may not be able to stay phased permanently – it may be like how whales have to surface every so often to breathe; they’d never get harpooned if they just stayed deep underwater, but it isn’t possible.
The Spore Spawn would be invincible if it never opened to reveal its vulnerable insides. Justified in that this creature probably isn’t sentient.
Mogenar from Metroid Prime 3 can only be damaged by using the Hyper Beam to destroy the sockets on his body… which means you have to blow up the red orbs inside the sockets first… of a boss whose tactical stupidity is resorting to the charge attack that lets you shoot the orb on its back.
The final boss of Metroid Prime 2: Echoes, Dark Samus (of course), combined this with an irritating caveat. Requiring the player to absorb the smaller Phazon attacks was understandable, and the Scan Visor even explained such. What it didn't say is that, in order to do this, you need to stand still - in an action-adventure/FPS hybridduring a boss fight. A timed boss fight that you'll automatically lose if you take too long to figure it out.
Justified in that, like the above example of Metroid Prime, Dark Samus (who was becoming increasingly unstable) likely needed to vent excess Phazon to prevent Phlebotinum Overload.
The final boss in Skyblazer. If he didn't try to punch you with his massive arm that you can use as a platform, you'd run out of magic far before he'd take enough damage to die.
Cortex in the first game. Why do you bother shooting at with green lasers that can be spun back at you, when you could simply have continued with the red and blue ones that couldn't? Also, lowering your energy shield at the same time you blow open a hole in the floor with mines is just asking for it, really...
That's not the end of it, although Cortex is the only one who purposely uses it. Koala Kong throws rocks at you, but he gets so tired that the third one lands in front of you, where you could spin it back. Then we have N. Brio, who eventually drinks a Super Serum that turns him into a powerful mutant. However, while attacking you, hitting the ground with his hands causes a block to fall from the ceiling, letting you jump on his head.
The Komodo Bros. in the second game. Had Moe just kept throwing his swords at Crash instead of sending Joe spinning around after him, the latter wouldn't get himself dizzy for Crash to send him bouncing back into Moe.
In the third game Tiny wouldn't be vulnerable if he kept trying to stomp on Crash instead of using his trident, Dingodile should've just kept shooting fireballs into the air instead of destroying the crystals around him, and is there any particular reason at all why N. Tropy creates a trail of platforms leading to him when he becomes exhausted?
Tropy's mistake from the third game is repeated again at the end of the Game Boy Advance game Crash Bandicoot 2: N-Tranced.
In Crash Twinsanity, Cortex not only uses three attacks, only one of which you can deflect, but he actually tells you how to beat him.
It's sort of like a birthday party, except... The exact opposite. Let's start handing out the presents... *gets on his hoverboard* This is from Tiny. *throws a bomb* This is from Dingodile. *throws another bomb* Ripper Roo, you shouldn't have! *another bomb* Pinstripe, how thoughtful! *and another* Oh dear, two of the same! *throws a final bomb and starts charging his ray gun* Here's one gift you can return! *fires*
The bandit king in Fable has an attack where he stabs the ground, usually when the player is standing right behind him. He's invulnerable at all other times.
Early on in Nicktoons Unite, you meet ghost prison guards who are intangible except when attacking. There's also similar fighters later on, but Danny gets the ability to turn intangible himself and is therefore able to hurt intangible opponents.
All four of the first bosses in Final Fantasies IV-VII are meant to get the player, who may be used to turn-based gameplay, used to the Active Time Battle system, especially the idea that the battle continues, and the boss continues to act, even if you don't, and that sometimes you'll need to wait.
The first boss of Final Fantasy VII, Guard Scorpion, has a defensive mode where it counterattacks any attack with severe force, but otherwise takes no action. Now ordinarily, it would make sense not to remain in this mode the entire time. However, the encounter begins shortly after the party sets a bomb to blow up the reactor they're in, and the defensive mode would be far more effective at delaying their escape.
Most notably, Asha, who can dodge absolutely anything, but thinks that dodging weak weapons is cowardly, even when he's about to die. Another boss, Iosa, is exposed only after you duck one of her attacks (granted, you only need to take advantage of this once) - which she uses, even though she also has attacks that don't leave her exposed. And the final boss, in order to charge up his most powerful attack, chucks incredibly powerful blasts at you... which can be reflected.
In an inversion, other assassins will use a move more frequently if they have successfully damaged you with it, and avoid moves that you hurt them with.
The fact that the final boss bothers fighting you at all is an example — at this point in the game, there's an Alpha Strike charging and only minutes away from being ready to fire, so if he really wanted to kill you, all he would have to do is leave. But then, killing you isn't what he really wants anyway. And if he didn't use that easily-reflected charge-blast attack, you probably wouldn't be able to beat him even so.
However, justified in the boss O2/Joy, who has to be deliberately provoked into doing his tactical suicide attack.
The Metal Queen in Brutal Legend, a giant spider who'll expose her soft underbelly periodically. Possibly justified in that a wild animal would take more than a few hits to be trained into not using that attack.
The Gray, which normally will harden in reaction to your attack. Its main attack, a steel-like tendril, leaves it unprotected until it is retracted.
The Horror, which is essentially the Gray on steroids — basically, a living buzzsaw that your sword bounces off of (leaving you wide open to being shredded). However, at certain points it fires out four flying saw blades, which again render it immobile and vulnerable until they return.
The Void Eater is only vulnerable when using its beam attack. Trying to attack it at any other point (or even getting close, for that matter) well result in you getting splattered all over the screen. If you attack a mobile black hole, you deserve exactly what you get.
The Razor Queen is only vulnerable to attack after it charges you
In the Sponge Bob Square Pants video games Battle for Bikini Bottom, The SpongeBob SquarePants Movie and Truth or Square there is at least one boss that does this. In BFBB every single boss does it except for the final boss's second form. Sandy Robot should stop using her Ground Pound move, and kick the player's ass with the "clothesline" move. And Patrick Robot is a real idiot in spinning around until he gets dizzy. King Neptune should stop using his easy to dodge sniper move. Also the giant Plankton in Creature from the Krusty Krab deserves a mention. He just stands there with his HANDS on the buildings waiting for you to hit them.
Nigel Block in 007: Agent Under Fire, both times. If he didn't drop his rocket launcher in the final battle all he'd have to do is float there with his jetpack until the withdrawing floor beneath you finally collapses and drops Bond into the pit. Earlier, he'd have been invulnerable if he hadn't called in the mooks that helpfully provide you with the rocket launcher when killed, nor would there have been a problem if he had chosen not to stop the fans protecting the air vents above the geothermal power station. On top of this, Bond never would have been able to locate and destroy the cloning facility had Bloch not shoved him into the pool that leads to it and instead shot Bond in the head while he was Distracted by the Sexy.
Most of the monsters on Monster Hunter series do this in some way or another. Either by exposing some weakpoint before (or after) their attacks or by sticking themselves into somewhere.
The final boss of Chip N Dale Rescue Rangers 2 drops bombs on the ground every time he flies into the air, which the player must then throw at him.
Actually, six out of eight bosses in this game are this, providing you with vital projectiles (and the other two are victims of Boss Arena Idiocy), but it's the final one that takes the medal. It's a bit better that his bombs are timed and have to be thrown at a precise moment... until you figure that if you do not touch the bomb at all, he probably will ignorantly land on top of it, letting it detonate directly under his bottom and hurt him. So, a somewhat skilled player can just dodge his attacks and watch the Final Bossliterally commit suicide, all by himself.
It would be easier to list An Untitled Story bosses that don't stop in their tracks and practically yell "HIT ME! HIT ME!". Well, there are two such bosses.
Two of the Bonus Bosses in the flash RPG Sonny do this. Omen has a move that makes both his next attack and yours much more deadly; if you already have a shield in place, you can hit him For Massive Damage while absorbing his attack. Dr. Herregods is even more egregious; he has one million hit points, and once you've knocked off a thousand or so he will heal himself for two million hit points. Too bad you have a move that reverses damage and healing.
Subverted by Zeus in God Of War 2. The fight has two different stages. First, Zeus is giant and you can't directly attack him. However, he summons sirens that release a deadly shockwave when you kill them, which damages him. The other stage is Zeus shrinking and fighting Kratos straight up. After awhile, Zeus remembers that he's the freakin' king of the gods, and starts bombarding Kratos with an endless stream of lightning. Kratos has to trick him to beat him.
Both C'thun and Yogg-Saron, the game's resident Eldritch Abominations, employ attacks that swallow or consume players, but also place them in a location where they can inflict damage on the normally invulnerable boss.
Icehowl in the Crusader's Coliseum, is a Bullfight Boss who literally charges into walls and stuns himself. Justified, as yetis aren't all that smart.
Hakkar the Blood God drains the life from the raiders, healing himself to nearly maximum health. However, all over the arena there are winged serpents that leave a cloud of poison when they die. If the players poison themselves, Hakkar drains the poison from their blood.
Baron Ashbury in Shadowfang Keep will occasionally start healing everyone, friend and foe, in the nearby area. To convince the players to let him channel it, he'll hit the entire party with an HP To One spell, necessitating a balancing act between staying alive and keeping him from undoing too much damage. If he had not healed himself, the players would have been in big trouble.
Ys 3: Oath in Felghana. If Chester didn't use That One Attack on you, you'd never kill him, as he is invincible until he uses that dash stab attack of his. He's got plenty of other attacks to murder you outright (especially the 2nd time you fight him...), he clearly doesn't need that Dash Stab, complete with the "HIT ME NOW" sound effect that also telegraphs he's about to do that move.
Similarly, Ernst in Ys VI will usually hit you first when you get close, but takes a moment to recover from his dash attack.
In Teenage Mutant Ninja Turtles: Turtles in Time, the stage four boss is Shredder himself manning a Spider Tank. The turtles can't hurt the mecha directly; they have to throw Foot soldiers at it. If Shredder stopped sending his mooks in, he would have all the time in the world to simply pick the turtles off with his guns. To be fair, at least the exact type of mooks you have to use as ammo changes with the difficulty level: easy has the normal variety, normal has the ones that throw shurikens and hard has the variety that block most of your attacks and can only thrown reliably by hitting them with a running tackle first, which can be tricky to connect with.
Almost all of the bosses in Billy Hatcher And The Giant Egg do this, with most of the bosses all you have to do is dodge everything until they mess up and become vulnerable, sometimes the bosses do the dumbest or just clumsiest things to allow them to be harmed.
Several bosses in Mega Man Legends are guilty of this - and most of them are piloted by, guess who, the Bonnes. Teisel's Marlwolf hurls powerful homing blasts, but every so often he'll turn away and open up a small hatch where the armor is thinner. His Servbots use it to lob hovering explosives; judging by Teisel's repeated howls of CLOSE THE HATCH!!, it just doesn't occur to them not to.
Both bosses in World 1 and World 2 of Fancy Pants Adventures suffer from this. The penguin is a Bullfight Boss who will always get his horns/flippers stuck in the wall, allowing you to jump on him. The rabbit, on the other hand, cannot be defeated by jumping on him at all, yet he still jumps high and drops spiders on you, though a snail always drops with them which can be used to beat him. To make matters worse, kicking him after hitting him with a snail shell will make him almost immediately use the same move again (with several more spiders, admittedly).
Though Golem's core in Bayonetta can be attacked any time the boss itself is attacking, only a few of its forms allow you to activate Witch Time. This is not a problem mostly, except during a timed challenge mission where the boss can only be hurt during Witch Time. During that mission, you are entirely at the mercy of fate whether it decides to use the two attacks that leave it vulnerable.
Both bosses of Iron Canopy in Darksiders are of this variety. The Brood Mother has a sucking in air "attack", which doesn't actually do anything but let you feed her a tasty morsel to give you a shot at her vulnerable backside. The boss of the area, Silitha, will occasionally interrupt her Teleport Spam to stand briefly still and taunt you.
Sardius in the SNES game Super Ghouls 'N' Ghosts; he can only hurt you with a Beam Spam attack from his mouth, and whenever he does so, he also creates magical platforms that raise Arthur up so he's level with Sardius' weak spot, his head.
The Brain Tank and Kochamara bosses have the most blatant cases of this. The former's only weak spot is its underside, which it reveals for a looong time when rearing up for a charge attack, and the latter has an attack that drains its shield into a weak beam-thing that you can reflect back or dodge while blasting him.
If the Butcher never did a single vertical slash his arm would never get stuck in the ground, giving a way to reachhis head. If Raz's imagined version of his dad didn't throw spiked flaming bowling pins at you during the second fight you'd have no way to damage the Butcher in the second battle.
The Star Wars Customizable Card Game features this for the second Death Star. A card called That Thing's Operational lets you move your Death Star and blast capital ships with it, but it also means that, when not around Endor or if the shield's not up, your Death Star is vulnerable. The light side player doesn't even have to play another card!
Parodied in The Simpsons Game, where the Simpsons discover they're in a video game and use this knowledge to their advantage. Unfortunately, Genre Savvy comes back to bite them in one battle against aliens, when Bart points out that the UFO fires four times and exposes its weak point. The aliens immediately start to wonder why they do that, and resolve not to do so anymore, making them completely invulnerable as they rampage through Springfield. The Simpsons get annoyed enough that they go confront God about it.
In the final fight of Mega Man Zero Dr. Weil can only be hit when his core is exposed to attack. If he waited the two minutes without attacking he'd be able to destroy Area Zero without Zero messing up his plans. Then again, he is a vengeful, hateful psychopath and touching down on Area Zero would destroy him too, so he's probably getting what little satisfaction he can get in seeing Zero die before him before he gets turned into a giant crater.
On the topic of Mega Man Zero we have the final boss of the first game, Copy X. During the second half when he uses his Seraph Armor, he destroys the majority of the floor with a set of spiked pillars that remain there for you to wall-jump off of. Had he not left them there, or at least moved them further out of the way, Zero wouldn't be able to hit him and Copy X could just laser-spam him to death.
Gate, one of the last bosses of Mega Man X6, is normally completely invincible to your attacks, but to attack you he throws energy balls that can have one of several different effects. When you destroy these, their pieces scatter in six directions, and they can hit Gate to damage him. At low enough health he will eventually start using a purple slash that can temporarily destroy the platforms you have to stand on, but he won't stop throwing energy balls. And to think, usually the series' fortress bosses are ordered from least to most interesting...
Nie R's Defense SystemGepetto, robotic guardian of the Junkyard, has the following attacks: fire a continuous laser stream from each fingertip, fire homing missiles, slam its floating hands on the Boss Arena, spawn endless robot drones with lightning and magic bullet attacks, a gigantic mouth laser, and... spawn bombs. Not armed bombs, just bombs. Which just sit there until you grab one and toss it. And it spawns them immediately before opening its mouth to charge up for the giant laser. Even Grimoire Weiss cracks wise at "such an obvious weak point."
The Ripper will sometimes stop attacking Dan in order to attack Kiya instead. This is the only time Dan can damage him.
When fighting Palethorn's Cruiser, the only time it can be damaged is when it flies past firing missiles, allowing Dan to get behind it and Attack Its Weak Point. If Palethorn only used his machinegun attack (which lets him stay at one end of the arena), he'd be unbeatable.
Some bosses will attack you with weapons that can be destroyed and converted into collectible bombs, even if bombs are the boss's only weakness, and without a constant supply you would quickly be left helpless. One boss attacks by constantly shooting floating barrels. It's almost impossible to be hit by these barrels, but they can be shot to yield health-pickups.
In addition, Star Fox 64 has a boss that is quick and manuverable. Since he's one of the few bosses that can be hurt without attacking a weak point, those are its greatest strengths. And every so often it flies to a platform in the middle of the arena and just stands there shooting at you, making it an easy target.
Spyro the Dragon is full of this. In the first game, Doctor Shemp would have lived if he didn't expose his back when he attacked, and Metalhead combines this with Boss Arena Idiocy; he could have easily stayed in the first area after you destroyed his power poles and would have been invincible. Second game, to defeat Crush, you need to attack him, then let him bring the ceiling on himself. It's somewhat subverted, as the rocks can hurt you also. Halfway subverted when he uses that against you, chasing you for a bit before smashing the ground. Third game, ALL THE BOSSES. The first four deserve special mention, as your ally helps you to hurt the boss. Of course the boss doesn't pay attention to the powerful kangaroo, the flying penguin, the yeti or the laser-gun armed monkey. The true final battle gives you a UFO, while the boss is just waiting for you to get on it. Mind you, the entire area is filled with purple acid, save for a small platform.
Metalhead may be somewhat justified in leaving the first area; since those poles are power poles, he may require them to operate at a combat-capable level.
The two battles with Red in A Hero's Tail are the embodiment of this trope. The first battle, Red repeatedly conjures up exploding boxes which almost never hurt Spyro and are the ONLY way to hurt Red. The second battle, he turns into Mecha Red, a robot version of himself, and decides that the best place to fight Spyro is in a room FULL of weaponry that he designed, all of which is capable of hurting him, and his former captive is outside in the control room making this weaponry available to Spyro CONSTANTLY.
The Lambent Berserker would be all but invincible if her ribcage didn't open and expose her glowing imulsion core whenever she charges, which she does a lot.
In the third game's DLC "RAAM's Shadow", General RAAM would be all but invulnerable if he would stop deploying his Kryll shield to attack the humans, instead relying on his blade (which the player can do during the segments where you play as him) or stalling until the Kryllstorm claimed them. In 1 (chronologically later), he smartens up and carries a Troika, which is why he is That One Boss.
In Treasure Adventure Game, several bosses suffer from this. However one robot subverts it: once you've attacked it twice when it exposes its weak spot, it changes its programming to not expose it any more.
The Ceaseless Discharge in Dark Souls can be easily baited into trying to leap at you over a huge pit. Attacking the arm it uses to keep from falling kills it immediately.
The Ninja Gaiden remakes usually make you work for your attack opportunities, but one notable example in the first game -Marbus- fights by flooding the area with some mooks and then taking potshots at you. While you can, if you're lucky, knock him out of the air on your own, he also tries to swoop down at you, invariably giving you a chance to beat the stuffing out of him.
The original Rayman game has Mr. Sax, who is defeated by having Rayman punch his musical notes back towards him, and Skops, who launches a homing projectile that hones in on Rayman's fists; time your punches just right, and they'll fly into him instead.
The only way to hit Sakit's first form in La-Mulana is when he extends his arm to the ground so you can climb on it and attack his face plate.
The second form has his face plate break, and a jet of flame burst from the arm otherwise used for climbing. Since the Keyblade is inside the Endless Corridor he's guarding, you're going to have to use shurikens to finish the job.
Played with in Hotline Miami. The second boss is a cleaver-wielding man who kills you instantly if you get close, but mostly keeps his distance. He occasionally lunges at you, which is easy to dodge, and eventually tosses his cleaver at you—it gets stuck in a wall, and he tries to pull it out while you're free to bash his skull open. But in the epilogue you get to relive the battle from the boss's perspective. When the protagonist comes in, you can kill him simply by walking up to him and ending his life with a single cleaver slash, as opposed to engaging in overtly complex and suicidal tactics.
King Toad in Guild Wars 2's Super Adventure Box: smack the jewel around its neck to loosen shards of gemstone; dodge its regular attacks until it's preparing to do its devastating tongue lash attack, during which time it will... just sort of stand there with its mouth open... and toss the gem down its throat, stunning the toad and enabling you to do damage. Rinse and repeat.
Subverted big-time by Mister Freeze in the sequel, who actually LEARNS FROM THE PLAYER'S TACTICS as the fight goes on.
Magic: The Gathering features a number of cards like this. Lich, Final Fortune (and its many reprints under different names), Nefarious Lich, Phage the Untouchable, Immortal Coil, and the Pact cycle will all kill you under the proper conditions. The trick is to win before those conditions are met.
That or use certain other cards in conjunction with those cards.
This is actually part of the color black's traditional color strategies. A large number of strong black cards demanded costs in addition to mana in order to activate or (more in line with the trope specifically) repeatedly harmed the player for having them out. This has waned over the years, mostly due to numerous ways to benefit from harming oneself or sacrificing creatures that have come along over.
The Omnidroid machines in The Incredibles can only be hurt by their own claws. That said, the new model is smart enough to not attack itself voluntarily.
Unfortunately, Real Life military tacticians and prizefighters do this sort of thing all the time. Examples will be omitted to avoid offense - think of your own favorite.