Pandora's Guardian, the giant, armored, demonic, fire-breathing zombie minotaur, fights Kratos in a narrow corridor with some sort of ballista mechanism at one end of it; Kratos can use the ballista bolts to chip away at its armor, and eventually defeats it by impaling it on the door at the other end of the room.
Clotho fights in an arena filled with deactivated traps that aren't at all suited for hurting someone the size of a normal human. Kratos, of course, uses them to kill her in an elaborate puzzle boss battle.
Perseus' preferred tactic is to use his helm of invisibility, then either sneak attack Kratos with his sword or take potshots with his sling. Unfortunately for him, he is forced to fight Kratos in a room with a shallow pool of water, meaning you can use the ripples and splashes to help figure out where he is.
In the third game Heracles is wearing armor made from the pelt of the Nemean Lion, whose golden fur is nearly impregnable. Heracles by extension would also be invincible, were it not for his tendency to stop, bellow loudly, and drop his guard with a bum rush long enough for Kratos to counter it and slam him into one of the walls of spikes set up around the arena that keeps him in place long enough for Kratos to remove his armor.
Xena: Warrior Princess has at least 3 such bosses. The first one is a Cyclops blocking your way. You are at a high cliff and the guy reaches from below, grabs the ledge with one hand and uses the other to smash you if you try to bypass him (why he doesn't try to climb up the cliff is beyond me). All attacks against him are useless. The solution? Keep slashing the hand grabbing the ledge and he'll fall off the cliff. Idiotic indeed. The second one is an invincible (to swords and chakram) fire breathing giant dragon that can only be killed by dropping stalactites on it. Admittedly, said stalactites are not unique to this level, you do get to make use of them to bypass obstacles in LATER levels, but still... The final boss is also invincible and constantly charging at you, and the only solution is to lure it to charge at the supporting columns. When enough columns are smashed, the ceiling will collapse and kill the boss while you hide under an uncollapsed section (if you have enough sense to rush there, that is.)
The fourth boss in Super Bomberman can only be damaged by being shoved into an electric fence with your bomb blasts.
Seasons: The Dodongo fight is like the original, but you need to use the Power Bracelet to throw him into the spikes after he swallows the bomb. Why are there spikes there? There's also the random giant metal spike ball in the area of the fifth boss, Digdogger. Good thing you got those Magnetic Gloves beforehand, and no, the spike ball doesn't reappear later. Both Agahnim and the Poe Sisters would have fared better if they didn't fight in areas with torches.
Ages: Smog averts this by making you play a 'game', changing the arena to try to get his separate parts together. The game, being more puzzle-orintated than Seasons, avoids this.
Gohma is invulnerable to grapple and sword attacks, because she is shielded by a thick layer of armour. Handily, the boss fight takes place in a cavern with a weak ceiling and a dragon's tail poking through it. Putting a little bit of weight on the tail causes part of the roof to crash down, destroying her armour.
Fighting the boss Jalhalla involves hitting him with light from holes in the ceiling and throwing him into spiked pillars around the room.
A memorable example is the eighth Colossus (Kuromori, the lizard), who you fight in a ruined Colosseum. He has incredibly powerful lightning attacks, and there is no way to damage him at first. If he stood still, he would be invincible. But, if you aggravate him and hide, you can get him to climb the walls to try to hunt you, at which point you can shoot him to make him fall down, leaving him vulnerable.
Both of the "dogs", smaller and faster Colossi that chase you around, are beaten this way. The first is said to be afraid of fire (the only one that is) and lives in an abandoned temple... that has torches around (the only place that has them, mind) so you can scare him with them to push him off a ledge and destroy the armour. The second chases you around as you bait it into knocking over pillars to you can hop onto a ledge, which it crashes into. The final pillars cause the ceiling to collapse onto it, destroying that armour as well. Had they traded locations at least the first one would have been impossible.
Also memorable is the 16th Colossus, Malus. His lightning attack is even more powerful and has strong knockback. Although he holds the high ground in his arena, the arena is also filled with defensive walls and tunnels you can hide in, to approach his blind spot.
In Super Metroid, Draygon's lair is surrounded by turrets that fire balls of plasma at intruders. A few missiles render them inoperable, however, leaving behind bare high-voltage circuitry that can be used to electrocute Draygon in four seconds flat. The boss can also be defeated the old-fashioned way with loads of missiles, but frying him is easier and considerably faster.
Crocomire would be completely invincible were it not for the conveniently placed lava pit behind him.
Subverted in the Mr. Freeze fight: Freeze didn't design his hijacked lair (an old GCPD forensics unit), but once you've used the environment to land a stealth attack on him, he promptly alters it via his suit's mechanisms so you can't use that trick again. You need to find about seven (ten in the New Game+) different methods of weakening him.
However, Clayface's boss fight resembles this. One of his attacks is to roll into Batman: if done right, you can have him roll into the explosives set in the corners of the room and severely weaken him.
The Queen in Ico saw fit to decorate her throne room with movable, mystical stone pillars that nullify her instant-petrification spell. All it takes is for Ico to drag them around the room and hide behind them at regular intervals until he can get close enough to take the sword she's vulnerable to and stab her with it.
Many Legacy of Kain bosses fill this trope perfectly. In Soul Reaver, Rahab's chamber consists of windows you have to break to shine sunlight on him, while Melchiah's chamber has retractible portcullises you have to lure him through to have them fall and hit his back. In Blood Omen 2, Faustus stands atop furnaces in a room full of mist, which you can use to become invisible and sneak up close enough to turn the furnaces on and burn him. In Defiance, Turel's chamber has four gongs that ring loudly and stun him when rung. The third boss of Blood Omen 2, Sebastian, isn't so much Boss Arena Idiocy as it is he's plain stupid — he crawls along the walls and leaps at you, and there's a laser in the center of the room. If you stand on the other side of the beam, Sebastian will leap into it to try and hit you, and hurt himself in the process.
Probably the best example from Soul Reaver is Dumah, who has spent so long in the spectral realm gathering strength that he is completely invulnerable to all of your attacks and would be unstoppable... if he didn't chase you into the blast furnace, or the one room in the fortress with something in it that can kill him. Weirdly, despite immolation being the only thing that can kill him, he's still completely immune to the fire and sunlight glyphs.
The first boss, Tiamat, would be unbeatable if her arena wasn't surrounded with torches and bomb flowers. The second boss, the Griever, is defeated with a freight rail carrier that War can punch into her gut. Straga's own weapon and the floor of his own arena are both used to kill him.
In the Griever's case, it's not technically her fault — she has no idea you can move something that big. Tiamat can allegedly be defeated using only your Crossblade without the bombs.
The Last Elephant-bot enemy fought in the Freakshow can't be damaged by Dan's normal weapons, but the cranes can be used to crush it by dropping spare parts on it.
The Count suffers from both this trope and Tactical Suicide Boss; the first stage of the fight requires you to use mirrors to reflect his spells back at him, and the second has you aligning them so that they reflect sunlight onto him, causing him to burn.
Ōkami has several bosses like this. The Spider Queen is completely immune to your attacks, her only weakness being her eyes inside her abdomen. But, oh, look! All around the arena are conveniently placed flowers that are here for no reason at all and that you can use to grab her hooks and open the Queen's abdomen, exposing her weak point! Kyuubi, Lechku & Nechku, and Yami's Boss Rooms are also full of lava, water, and thunder without which you wouldn't be able to do a thing to them.
The boss of the theater level in Psychonauts is slightly smarter than most in that he destroys the spotlights once you've used them to stun him. He's still not smart enough for basic pattern recognition, and ignores the other identical spotlights who are all within spitting distance of each other. Even after you successfully used the first two on him, he doesn't think to destroy the third.
A number of bosses in the LEGO movie crossover games suffer from this, too; and at least one of which, the Rancor, is also such in the movie where it came from. At least in the movie, it's Justified as being put there by its owner, Jabba the Hutt.
The first act-boss battle is against a three-headed hydra out in a large courtyard. The Hydra heads' breath is quite deadly. However, it just so happens that there was a convenient piece of thick wall right in front of the Hydra's position, which your characters can hide behind while recharging their powers.
The first part of the battle against the dragon in The Northern Reaches has the dragon alternating between aerial attacks and long range attacks from various perches around the rampart you're standing on. There also happens to be a large ballista that can be turned toward said perches which is the only way to damage the dragon at this stage. Partially Justified as the tower was set up for defense, not as the dragon's lair.
A straighter example is the fight with Valdis at the top of Zaramoth's Horns. Near the four corners of the room are "Eyes of Zaramoth", small statues with wings and a single eye that will fire a very powerful beam shortly after being struck. Naturally, this is the only way to hurt Valdis at this stage. The fight concludes with a giant version being gradually uncovered that Valdis must be positioned in front of.
The Beat 'em Up was chock full of such examples, but the most blatant is the penultimate boss fight of the game against the symbiote Carnage. As Spidey tells you, a symbiote is weak against fire and supersonic frequencies, so naturally the room you fight Carnage in has a massive sonic bubble in the middle (presumably built by Dr. Octopus as a failsafe against Carnage) for you to knock him into. Other such examples include the fight against Doc Ock (with convenient "disable forcefield" buttons lowered to the arena one-by-one) and the Rhino boss fight (with electric pistons for the Rhino to charge into and electrocute himself with). The sequel lampshaded this, even, with the final battle against Hyper Electro where, after you trick the pure energy final boss into zapping one of the several generators in the area, which sucks his energy away and makes him physical again so he can be attacked, Spidey quips "Man, you'd think a guy would learn after a while!", or with "Remember kids, good always wins because Evil Is Dumb!" This turns against you if you didn't destroy the electric tower, which will refill all of Electro's HP. Being that it takes up half the area, you should feel REALLY dumb.
Also in the sequel, the second fight with Sandman (the first being impossible to win) takes place in a construction site, which seems logical until you spot the open plumbing.
There are a few fire bosses in Dynamite Dux, which can only be harmed by water guns, of which there are two placed immediately before each encounter.
Mortal Kombat Deception features Onaga, who is stunned when you destroy (touch) one of six objects called kamidogu, which are the source of his power. Naturally, he places them along the fringes of the arena, spaced 60 degrees from each other.
First Person Shooter
Quake has at least two of these. The first chapter boss is completely immune to all damage apart from two adjustable columns that can shoot lightning between them. The final boss is impervious to everything except a floaty teleporty doohickey. Neither of these unique architectural features can be found anywhere else in the game.
Aaron wields a chain that he uses as a whip against you. The only way to defeat him is to position yourself in front of one of the rings on the wall and dodge his attack. His chain will get trapped in the ring, and you can attack him while he's busy trying to get the chain out.
Ambrose is invulnerable because he grabs your magic Gel'ziabar stone and puts it in his axe. He's only killable when a giant Gel'ziabar dog comes out of nowhere and attacks him, and then you can only kill him by first shooting the stone out of his axe. If you don't rush up and kill him right away, the dog will vanish, and he will pick up the stone, put it back in his axe, and resume being invulnerable until the next dog attack.
The Nihilanth in Half-Life is invincible by drawing upon the power of (and expending) energy orbs floating around its head, which get replenished by easily destroyable crystals on the walls of its chamber. Once the crystals are gone, the orbs eventually run out and the Nihilanth is toast.
The final boss of the sequel to Doom is a giant invulnerable face in the wall of an arena which spawns endless monsters from the hole in its forehead; its only weak spot. You can shoot rockets into the hole to kill it, but such a task would be impossible if not for the arena providing a convenient elevator that reaches the height of the brain hole.
Hack And Slash
In Devil May Cry, the battle with Nightmare. He first appears as a completely invulnerable pool of black goo which swallows Dante up if he touches it. Hitting the switches that surround the arena enough causes him to take on a more solid form which, while more dangerous, is also vulnerable after it attacks.
Apparently, those were installed to control Nightmare and make it do what the maker's wanted. Considering how powerful Nightmare was, it's understandable. Possibly, they actually PREFER Nightmare in "Battle" Mode as the alternative doesn't really...do anything.
Lyran, a Lich in Baldur’s Gate: Dark Alliance 2. The first time you meet him, he's completely invincible, and you're forced to fight your way through his castle while keeping your distance from him and dodging his spells. Eventually, you find the ossuary where his one weakness (his mortal remains from when he was alive) is stored, and he decides to lock you in the one room where he's vulnerable so he can do battle with you properly... Needless to say, it goes poorly for him.
The scarecrow boss in the Plague Town level in Gauntlet: Seven Sorrows seems to have left a number of giant pots of flaming oil sitting around so you can tip them over on him.
In Armored Core 3, the higher ranked Arena opponents either frequently use their Overboost or take to the skies and will keep boosting upward even when they hit the ceiling which usually gave you two easy ways to kill them. One was in the desert arena where you could trick the OB-saavy opponents to dash right out of bounds which counts as an automatic forfeit. Or you could go to the parking lot where their movement will be limited to a few feet in the air so you can pound them to death with the strongest weapons and they'll just float there and take it all.
It would be nearly impossible to defeat Anub'arak in the Crusader's Coliseum save for the conveniently placed frost orbs floating around his lair which, when knocked down, form icy patches on the floor that prevent his minions from burrowing and stun him when he runs into them.
Karsh Steelbender in Blackrock Caverns is almost invulnerable to player damage unless he's lured into the molten metal conveniently pouring down out of his forge. Of course, doing this also causes him to inflict massive fire damage to the entire party.
In Naxxramas Instructor Razuvious hits hard enough to make him nigh-untankable. However, his nearby students can be mind-controlled by priests and sent to soak up his hits. It's even worse in the 10-man version, where for no particular reason (save for the fact that a 10-man raid may not have people who can mind control) there are mind-control orbs sitting right there that anyone can use. That's just asking for trouble.
Professor Putricide in Icecrown Citadel would kill the entire party with pools of ever-growing slime if he did not conveniently have a bottle on his desk that transforms the person who drinks it into a mutant monster that eats this slime.
The Ulduar Flame Leviathan and indeed the entire section leading up to it would be pretty much impossible if a bunch of siege engines hadn't been left right by the entrance. The Flame Leviathan would also be much harder if there weren't barrels of pyrite floating all over that can be shot down and it tricked into running over or shot at it by the vehicles.
Brann and the other allies who show up built the siege engines from salvaged Titan technology.
The blind dragon Atramedes of Blackwing Descent makes the mistake of fighting your group in a room where the walls are lined with shields that can be smacked like gongs to deafen and disorient him. It goes without saying that this is the only way to beat him.
Also from Blackwing Descent, Chimaeron is a boss that deals massive amounts of damage for his level -Massive enough that the fight would be impossible if not for the Bile-O-Tron, a device that prevents Chimaeron's attacks from killing anyone who has more than 10,000 health. This case is partially justified in that the Bile-O-Tron was built by a gnome Nefarian was keeping prisoner and was not supposed to be there, but then, one has to wonder how Finkle managed to build it while caged in the first place (or sneak it in when he was captured, if he'd done so already).
Magtheridon has five cubes that must be used to banish him when he uses Blast Nova. Justified in that Illidan's forces were keeping him prisoner there.
Razorgore the Untamed has a mind control device in his room, which you need to use to make him destroy the eggs before you can defeat him.
Bloodlord Mandokir is able to instantly kill any player with almost all of his attacks. Good thing he chooses to fight you in an arena surrounded by benevolent spirits who can resurrect you and make you stronger.
Morchok from Dragon Soul is another example. He summons huge earthen spikes right before casting Black Blood of the Earth. The spikes themselves don't serve any purpose except for the players to hide behind to completely avoid damage from Black Blood of the Earth, which, if stood in, will kill you after a fairly short amount of time. Basically, he'd easily wipe the group if he didn't also summon those spikes for no apparent reason. In a similar vein, the final boss of Sethekk Halls charges up a massive Arcane Explosion that does fatal damage, but is easily blocked by hiding behind one of the several pillars he has in his room.
The Spine of Deathwing fight takes place with the entire raid riding on Deathwing's back. At various points in the fight, he will attempt to shake the players off by rolling. This is potentially a One-Hit Kill on the entire raid, but it can be avoided by using the numerous tentacles sprouting from Deathwing to secure yourself to his back. Though Deathwing didn't put the tentacles there intentionally (the sheer amount of power he's absorbed is causing his body to mutate and collapse) one has to wonder why he does not simply slam his back into the ocean he's flying over and crush/drown the players.
Spiral Knights: The Snarbolax is invincible unless the Beast Bell in his boss room is rung when he's close enough for it to stun him. The Roarmulus Twins are immune to Knights' weapons, but not their own missiles; the Twins are placed directly across from each other with switchable walls between them.
The Undead Lich in Guild Wars Prophecies chooses to fight you in the one place where his immortality can be stolen. On a lesser note, he doesn't even bother stepping away from the nearby lava, sometimes even teleporting into the lava, causing him to burst into flames.
Scarlet Briar's Hologram fight in Guild Wars 2 is only possible in the first two phases due to it creating energy puddles that allow you to do full damage. In the third phase these disappear, but are replaced by small holograms that are much easier to kill, damaging the main hologram.
Practically every fight against Bowser in platformers is like this. The most blatant example is in Super Mario 64, where Bowser has learned to recover from the bottomless pits/lava that usually defeated him in the past and would be completely invincible if it weren't for the floating mines surrounding his arena that Mario has to go suicidally out of his way to be hurt by.
Happens quite a bit with the other bosses in the Mario series. For example, the electric fence in the arena where you fight Topmaniac in Super Mario Galaxy, lots of examples in the final Bowser battle (plants and those blue panels with lava underneath), etc.
In the Wario Land spinoff series, a certain few bosses have lava around the arena they need to be smashed into, and another has water on both sides of the arena for no real reason than the boss to be knocked back into.
In the the first Super Mario Bros., Bowser (and each of the fakes) stands on a bridge. On the other side of the bridge is an axe. If Mario can pass Bowser and get the axe, the bridge and Bowser fall and you win. Classic.
Even worse in New Super Mario Bros. where the fight is the same, but the axe is replaced by a large button with a skull on it.
The "sequel" has this happen to Bowser Jr. in all three of his boss battles. The first fight has him stay far above you... while there are propeller blocks for you to use to jump on or throw at him. The second has him giving you your own Clown Carto play Bumper Cars with him (as well as electrified walls). The third has him jacking his dad's own Clown Car and throwing giant spiked bombs at you... while the floor is extremely rickety and can be used to send the bombs back to the kid. Yeah, stupidity clearly runs in the family. On an additional note, this also happens to Wendy in her second fight with the water levels lowering after a while, though this can be argued as being Kamek's fault for making it like that.
Inverted in the very next Giant Bowser fight where you, as Bowser, have to beat the enemy before you get led onto a collapsing bridge over a bottomless pit (which gives you an instant Game Over if you reach it).
In Mario & Luigi: Dream Team', the giant bosses have some of this going on in their arenas. Earthwake would be pretty much invincible if it didn't conveniently fight near the Dreamy Wakeport habour, leading to scenes like those in the battle with the Tower of Yikk from Bowser's Inside Story (aka you get to knock it in and smash the hell out of its head with a hammer). And Giant Bowser would be invincible at the end of his battle had the fight not took place in a lava filled wasteland that he could be easily knocked back into.
In the Western Super Mario Bros. 2, Wart can only be killed by having vegetables thrown in his mouth. He has a machine in his throne room that shoots vegetables. He also opens his mouth a lot. Wart is not very smart. And apparently there's a reason. Wart stole the Nightmare Machine and reprogrammed it to bolster his army. Problem is, the thing has a mind of its own and it decides to help kill Wart.
King Totomesu, the first boss of Super Mario Land, has a boss stage that looks just like Bowser's, and can be defeated in the same manner - either run under him when he jumps so that you can hit the button behind him, or pelt him with five Superballs.
Bowser was actually like this since he was a kid. In the final boss fight against him in Yoshi's Island, Baby Bowser fights on a rubber floor that can send shockwaves around the entire floor when Ground Pounded on, which both him and Yoshi's use to fight each other. In the second phase, he's ridiculously huge and charging at you from the background as an artificial time limit, and he'd be unstoppable if seagulls didn't pop up now and then to deliver giant eggs for you to use on him.
Tap-Tap the Red Nose probably passed down the idea; he's on top of breakable blocks over some lava.
In Super Mario World, Bowser's defeated by having Mecha Koopas thrown at him which he himself deploys. It's not like those are his only means of defense against Mario in this battle. He also causes fireballs to rain down, attack Mario directly, and throw a giant black ball at him.
Super Mario 3D World has a downright silly example. After a confrontation with Bowser, the top of his tower contains a massive POW block. Which he's standing on!
Even before that was Donkey Kong who would finish the fight in an arena where Mario could remove the pegs, causing Donkey Kong to fall down.
This lives on in the Mario vs. Donkey Kong series. An especially egregious example is in Mini Land Mayhem in the fourth level how Donkey Kong hits the button to mix up the pipes. On the surface it seems clever, but if he NEVER hits that button, then it's impossible for a mini to ever find their way to one of the bombs needed to take him out.
In Sonic Unleashed, the boss Dark Gaia Phoenix fights you in an area with the game's first throw-ready water barrels, which are necessary to hurt it. Said barrels continue to show up solely in areas where fire-protected enemies attack you.
In Sonic Adventure, Amy's story has her pursued by a robot, Zero, who can be stunned by her hammer but never actually destroyed, and if you hit it enough times, it becomes invincible. However, in Amy's only Boss Battle, the arena is surrounded by nodes between which electricity constantly arcs. Knocking Zero into the electricity stuns it and exposes a vulnerable button which may be hit by Amy, thus damaging it.
Chaos 6 from the same game would probably wear down Sonic and Knuckles if those robots didn't come into play; stunning the robots then causing Chaos to ingest them freezes him, thus allowing attacks to actually hurt him. In Sonic's version of the battle, Eggman himself dispenses the robots, then complains when you're able to use the robots against Chaos. And, when you damage Chaos/Eggman after that display of idiocy, his reaction is a memorable "No way, I can't believe this!". This man has an IQ of 300.
And let's not forget the very first game; the Green Hill Zone boss would be unbeatable if he didn't decide to strike in a screen with floating platforms.
And ditto for the Star Light Zone boss, which would be impossible to reach or harm if it didn't drop its bombs.
The Biolizard would be a lot more difficult to kill (not that it already isn't), if it didn't produce a load of pink bubbles, that could be used to reach its life-support system. Stopping to catch its breath is more forgivable, since if it didn't, it would, well, die.
Easily the biggest example from the Mega Drive era would be the Sandopolis miniboss from Sonic and Knuckles, which is completely invincible to the heroes' attacks but definitely not invincible to the pit of quicksand you can lead or knock it into.
From the same game, the "normal" boss of Death Egg (before the final boss sequence) automatically deflects all attacks and drops robotic minions on you, which are the only things that can damage it. The only reason you can even do so is because he picked a room that happens to let you flip the gravity.
Shadow the Hedgehog isn't immune either. Dr. Eggman's "latest and greatest invention" includes buttons that couldn't possibly benefit him, most notably the ones that trigger Shadow Fever. It may have been used in order to power up those Shadow androids he made, but he really should have had the foresight to disable that feature when the real Shadow showed up.
Sonic Adventure 2 has Egg Golem, a boss with a big power source on its head just perfect for being attacked by a tiny blue hedgehog. Hoever, Eggman had the smarts to make the robot much too big for Sonic to easily climb to the top of. Except he also decided to put stepping stones up the side of the robot for Sonic to climb. With pictures of Sonic on them.
It starts with this as early as the first boss. The turrets that pop up on occasion do a lot of damage to the boss, with the added incentive that you want to deal with them quickly to prevent them from shooting you. The Komato Sentinel would probably wipe the floor with Iji if not for the conveniently located electrified pads she can knock it into, and the initial form of Iosa The Invincible is very hard to kill without using the lasers in her chamber. Enforced because Iji has an experience system that lets you build combat abilities or more stealth/exploration oriented skills. Thus, to prevent players taking a noncombative route from getting stonewalled, nearly every boss is a Puzzle Boss.
Defeating Proxima, the Sentinel without using the electropads actually gives you a reward. It's not too hard provided you brought enough ammo. Iji comments on it by saying "Guess it pays to be prepared".
In the first game, Ripper Roo is invincible. The only way to harm him is to jump on TNT that's moving down the river between the two platforms and hope that the explosion hurts him. Same thing happens in the second game, but this time he's a Tactical Suicide Boss and the whole thing is just surviving long enough for him to kill himself.
In Crash Bandicoot 2: Cortex Strikes Back, Tiny is invincible. The only way to harm him is to hop around on the platforms, avoiding him, until they blink red. Once they do, hop onto a non-blinking platform and hope that Tiny lands on a blinking platform and falls, hurting him.
In Crash Bandicoot 3: Warped, N. Tropy is... not invincible. However, he has you at the far end of the arena from him, and thus you can't do a thing, not having a distance attack. Until he switches the platforms to create a direct trail to him... and then takes that moment to catch his breath and stop attacking you.
In Crash Bandicoot 2: N-Tranced, Fake Crash copies all of your movements and will kill you instantly if the two of you touch, so you have to lead him into the traps on his side without also getting maimed by the traps on yours.
In Crash Mind Over Mutant, the boss fight against Evil Crunch and N. Brio has the boss in a decrepit weapons factory - next to a conveyor belt steadily supplying TNT Crates, which are the only thing capable of blowing up Crunch's cannons. Then, later on, when facing a mutated Cortex, the boss is defeated by taking control of Cortex and spinning a screw out of the ground then body slamming the self-destruct switch for the space station the fight occurs on. The screw in the ground is only uncovered when Cortex's defenses and minions are defeated.
In Sly Cooper and the Thievius Raccoonus, Muggshot's arena is filled with person-sized grounded light bulbs. To defeat the boss, you have to reflect light onto the light bulbs (thus making them completely impractical for non-combat use), and lighting them all hurts Muggshot and melts his guns, despite not hurting your character. There's no attempt whatsoever to explain why Muggshot keeps these things in his office.
In the first game in the series, you have to twice fight Necky, a boss that only can be hurt by jumping on its head. Unfortunately, its head is too high to Donkey or even Diddy reach. Good luck that someone unexplainably left a tire lying around, right?
The bosses in Donkey Kong Country Returns were generally good about choosing locations that weren't potentially lethal towards them if utilized properly by Donkey and Diddy. The one glaringly obvious exception was Mangoruby, whose boss chamber contained three wheels with switches on them that, when pounded, would de-electrify Mangoruby's body and allow Donkey to Goomba Stomp her.
Most of the boss battles in Donkey Kong 64 have a random, convenient, COMPLETELY unnecessary exploding barrel in the middle of the room that, if removed, would make even the first boss impossible to defeat.
In Mega Man & Bass' Burner Man decides to fight the titular heroes in an arena blocked off by two pits of spikes (which usually are a One-Hit Kill, but will simply damage Burner Man for about 1/4 his total health). Although Burner Man himself is at least smart enough to avoid the pits while he's dashing around and trying to hit you, if the player brought along Cold Wall (Burner Man's Kryptonite Factor), and slides it into him...
Cut Man, from Mega Man 1, is weak against Guts Man's power. Guts Man's power is simply being very, very strong. Mega Man can use this strength to lift giant brick blocks and throw them away or at enemies. And for a literal case of Elemental Rock-Paper-Scissors, guess who's got a few brick blocks in his arena for no particular reason?
In Spyro 2: Ripto's Rage!, the boss Crush combines this with Tactical Suicide Boss. He fights you in an arena where using his giant club sends debris falling on him from the ceiling, the only thing that injures him. He has plenty of other attacks as well, but if you annoy him he'll keep using his club.
In Attack of the Rhynocs, Ripto is defeated by leading him into destroying the three statues throughout his throne room.
No boss in the Spyro universe compares to Red in this trope. You fight him twice - the first time, he seems at first to be competent by magicking up things that only hurt Spyro - and then for some reason he makes exploding boxes appear, which you push into him to take down his health. Despite the fact that they're the ONLY things that can hurt him, he makes these appear all throughout the battle. Then you fight Mecha-Red, a robotic version of Red, and he 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. And the award for Most Idiotic Boss Ever goes to....
The final boss of Prince of Persia: The Two Thrones, Vizier can manipulate matter telekinetically and at the first stage of the battle he tosses debris at you which is perfectly understandable. Then he makes debris revolve around the arena trying to run you over with them which is also perfectly understandable. Finally he soars high into the air where he would be completely unreachable for you... if it wasn't for the debris that he, completely unintentionally, of course, arranges as a contrived obstacle course.
Earlier on, the first boss in that game, the giant, is fought in an arena with walls that can be climbed to leap at the boss and stab it in the eye. Even after you do this the first time, it doesn't smash anything that could allow the prince to repeat the process with its' other eye. Granted, he's not the sharpest tool in the shed.
In Prince of Persia (2008), the Warrior would be completely invulnerable, were it not for some well-placed construction pillars in the boss arena.
Rayman 3: Hoodlum Havoc has the boss battle with Reflux. Reflux is a Knaaren. Knaaren are an invincible race — seriously, you have to get through an entire cave of them, and none of your attacks do a thing. Avoiding them is the only option. So Reflux, apparently one of the most powerful Knaaren (never having been defeated before, according to the Knaaren leader) should be a Hopeless Boss Fight, shouldn't it? Nope. Reflux uses a staff as a weapon and you attack that to defeat him.
In the first Rayman Mr. Stone can't be hurt at all by your attacks, and even the smaller rock men are Invincible Minor Minions that stay down only for a few seconds. Unfortunately for him the arena features a tall idol tower that can be knocked over his head and smash him to bits.
The Medusa boss in Hercules can only be killed by running around and hiding behind shields mounted on the wall to reflect its gaze.
The boss of World 2-S1 (Atlantic Abyss) in Super Mario Fusion Revival is a White Shark. It is invincible against all of your attacks, it constantly seeks you out, and it rushes you at blinding speed. There are large jellyfish that enter the playfield occasionally. The only way to damage the White Shark is to dupe it into rushing into a jellyfish.
The boss of the Chateau du Vent has plenty of throw blocks in its arena. If they were replaced with an ordinary platform, the boss fight would be impossible.
The boss of Chateau de la Terre still has the throw block problem, but there's only three throw blocks in said arena.
Lampshaded in Portal: the only way you can defeat the rogue computer GLaDOS is by manipulating incoming rocket fire through portals to hit her. Though GlaDOS has so far been shown to have control over all the systems in the entire facility, she notes that her morality core, which you destroyed, "must have had some ancillary responsibilities" and thus she is unable to deactivate the turret.
It initially averts this; GLaDOS has learned from the last time and presents you instead with what would be a completely lethal trap, if you and Wheatley hadn't sabotaged her turret and neurotoxin production during your earlier romp through the facility.
Likewise, although Wheatley claims to have studied footage of GLaDOS' defeat and doesn't make the same mistakes, he makes entirely different mistakes instead, building his "lair" near convenient gel tubes and hurling bombs at you instead of waiting for the neurotoxin to do its work. Mostly justified though, as he has been well established as being deliberately made deeply, deeply stupid. Plus, he wants the satisfaction of killing you before the entire place explodes. It's also subverted in that he does have one final surprise in store if you beat him, perhaps his triumphant moment of Genre Savviness, and it's not his fault that it doesn't work. He also tries turning the bombs off after you hit him the first time, but he's apparently too damaged to do so, like GLaDOS above.
The room you fight in has no viable portal surfaces (your only method of attack) before Conversion Gel tubes start bursting, so his plan was almost airtight. Unfortunately, somebody was stupid enough to try throwing bombs at someone standing behind a Conversion Gel tube, causing it to spray everywhere. After the sprinkler system activates and washes away the gel he even comments on how he should have triggered it himself if only he had known it would work (or been able to turn them on, given his state at the time).
Angry Birds generally has no bosses,(excepting the Space version) however, in its Cross Over with Rio, we are treated to two bosses from that movie. These bosses are surrounded by explosives and rocks that allow the player to inflict even more damage than the birds ever could.
Of course, in every other instance in the Angry Birds games, the pigs always place more fragile materials such as glass or wood where it would be much smarter to place stone or pillars. In some cases there are even TNT crates within their structures, or, as said above, boulders to bulldoze fragile structures or unfortunately-placed pigs.
Role Playing Game
Super Mario RPG plays service to this trend in the fight against Bowser at the beginning of the game: you fight on chandeliers held up by Chain Chomps, so instead of attacking Bowser, you can just attack the Chain Chomp (though oddly averted in the fight against Boomer on the same chandeliers later, now held up by Shy Guys. You can't attack the Shy Guys, though Boomer attacks his own Shy Guy after losing.
BlastMan in Mega Man Battle Network 6 has two metal cubes on the field when you fight him. Those become important, when he uses his strongest attack, sending a wave of fire from either up, down or side and forcing you to hide behind the cube. Later, when you re-match him, if you play Falzar version, you can suck in the cubes using Dust Cross, and fire them off at BlastMan for nice damage. Doing so will leave you defenseless when he uses his wave attack, though.
BubbleMan in Battle Network 3 subverts this. He spends the entire battle with a bubble shield, and behind a rock. The bubble shield will take one hit regardless of damage, the rock takes some time to destroy, and his attacks are designed to keep you busy.
A short parody "Dr Wiley Fails at life" inverts this, causing Mega Man to run into a room full of instant death spikes and die.
In Kingdom Hearts, you fight Hercules in an arena you've already used for dozens of battles. Every single time before and after, it was completely empty, just a flat ring. For the Hercules fight, it's filled with barrels that you need to get rid of the hero aura which makes him invincible. The barrels being provided may be justified by the fact that it's more of a friendly practice match than anything else and Hercules is giving Sora sporting chance. There is, however, no explanation for why a wooden barrel is able to weaken Hercules so much.
Dagoth Ur of Morrowind fights you (after a short introduction where he grants you the first blow) in a room that also houses the source of his power, the destruction of which would make him mortal. Granted, he wouldn't have reason to expect you to be willing (or even know how) to destroy that source, and he has to hang around the place anyhow to keep the Tribunal from sneaking in and renewing their power...
The Dragon God in Demons Souls would be unbeatable...if it weren't seated right between two huge ballistas pointed at its shoulders, put there by the Burrowers in case it ever ressurected.
Pokémon Red and Blue and their remakes have something similar to this. Lt.Surge is an electric-type gym leader. Yet he built his gym in a city with a cave full of Diglett and Dugtrio, ground-types that are immune to electric attacks. He can be tough to beat, but catch a couple Diglett before going in,keep spamming Dig, and you'll breeze through.
The Pokemon gym leaders make a bad habit of this. Skyla keeps her gym right outside a cave full of electric types (and if you someone managed to avoid catching a Joltik, Tynamo, or Boldore in Chargestone Cave, she makes you to go north on Route 7, where high-level Zebstrika hang out). Blaine's gym is on Cinnabar Island—you need a Pokemon that knows Surf, a powerful Water-type move, to even get there in the first place. And an NPC standing outside Roxie's gym flat-out tells you that Virbank Complex, just a few steps to the south, has Magnemite: a Steel-type that's completely immune to Poison-types.
Dark Cloud has a partial example of sorts, mixed with Tactical Suicide Boss: Around the boss arena in the Moon level, there are barrels of an unspecified liquid lying around, which the boss (a minotaur) will attempt to drink. When he does, you can shoot them with Osmond, causing them to explode and damage the boss. However, the "idiocy" and "suicide" elements of this are downplayed by the fact that, should you fail to do so, the liquid has the same effect as a Stamina Drink, causing the boss to inflict double damage and take half damage.
Stealth Based Game
In the original Metal Gear Solid, Liquid Snake can't be defeated through normal means and you don't have access to weapons while fighting him. Conveniently, despite having hours to prepare for the battle, he chose to stage the fight on top of a Metal Gear with nothing to prevent him falling off. Not that falling off stops him for long. Possibly justified, since his entire plan had basically already been ruined and airplanes were coming to bomb the place; he might have just wanted to make a dramatic last stand. Also, Snake could fall off the Metal Gear too, so it's not lethal to just him.
How about the Hind-D battle? He decides to blow up the whatever-they-are-tanks on the roof with his last missile... after his helicopter is already going down. Had he done that much sooner, he'd have taken out the only cover you had and you'd be at his mercy.
The Pain from Metal Gear Solid 3. He's a guy who's covered in bees. The only place where you can't be hurt by his insect 'brothers' (though all worker bees are females without the reproductive capabilities of the queen) is in water. So where does The Pain, in his infinite wisdom, choose to ambush you? On a rock in a lake, of course! How could that possibly backfire?
From the same game, The Fear. There are two massive mistakes he made in choosing his ambush spot: first, it's the same place he tested his poisons, thus the rabbits there drop antidotes. Secondly, the place is littered with poison dart frogs. Given that The Fear will eat any food in the arena when his stamina drops, shooting a frog makes the fight even easier since they will poison him, dropping his stamina further.
In Beyond Good & Evil there are some Alpha Section guards who, unlike the rest, don't have an air tank (thus, they are immune to the usual tactic for disabling them). Whenever you face them, they either can be sneaked past, or they are standing next to a moving platform that can be activated to send them to their doom. The Alpha bases also contain military robots that cannot be destroyed using conventional attacks; however, they can always be pushed into an electric barrier conveniently placed nearby, destroying both the robot and the barrier.
In The Chronicles of Riddick: Escape from Butcher Bay, Riddick has to fight a Humongous Mecha at one point to advance. You fight him in an open, well-lit room, in a game that heavily emphasizes stealth, and your firearms barely scratch its armor. Good thing there are explosive fuel canisters constantly moving around the room.
The T-078 Tyrant in Resident Evil: Code: Veronica has a distinct advantage in the enclosed space of the plane, only to be defeated thanks to a lone crate in the cargo hold.
In Resident Evil 4 Salazar's Right Hand (i.e. one of his two main servants) has several liquid nitrogen tanks that slow him down and make him vulnerable in his area, and when you fight two El Gigantes there's a lava trap you can use to kill one, but it technically isn't required in any way.
In Resident Evil 5, the second boss battle against Wesker. It's oddly convenient that he can't see you in the dark because of his sunglasses (which he won't take off no matter what), and you just so happen to be fighting him in an arena with easily-located light switches!
In Luigi's Mansion, Boolossus is completely invulnerable to everything except being pulled by Luigi's vacuum attacks. Conveniently, the balcony where you fight him inexplicably has a pair of stone unicorns. The trick to beating him is to use the vacuum cleaner to pull him onto the horn of one of the unicorns, 'popping' him into the smaller, vulnerable Boos which make Boolossus.
It gets better. The Boos themselves are far too swift to vacuum up, and can't be stunned... ...but they CAN be frozen by the ice element, and the aforementioned two stone unicorns are both iced over and inhabited by elemental ghosts during the battle.
In Luigi's Mansion: Dark Moon, the Tough Possessor would be entirely invincible with its suit of armour possessing shenanigans and stuff... except the room just happens to have two infinite respawning carpets on the floor that you can suck away to trip the knights up with. And when it possesses a giant suit of armour... the two carpets are perfectly placed to trip the giant knight up in one go.
The Big Boo in Treacherous Mansion has this too, in much the same way as Boolossus in the last game. It can be broken apart by spiky objects... and there's a giant train with a drill on the front going around a track nearby for you to fling the boss at. And despite the Boos being unable to be caught in the Poltergust, said train just happens to have a bunch of circus animal style cages lying open as the the rest of its carriages, providing a nice convenient place to trap the ghosts in.
In Silent Hill 4, Walter and the "god" he summons are completely invincible... except for the spears that spawn in the very place where you fight him.
Third Person Shooter
In Psi-Ops: The Mindgate Conspiracy, Marlena Kessler fights you in a room full of liquid nitrogen canisters. This is rare as there's an actual reason for her to fight you there; her fusion cannon wasn't exactly portable. Although the infinitely replenishing liquid nitrogen canisters aren't so excusable. Notably, it's also possible to fight her without the cannisters after you destroy the cannons (though it's insanely difficult).
In Ratchet & Clank, the Snagglebeast has a shield which can deflect any type of projectile. The only thing that can hurt it while the shield is up is falling into lava. Not only is the arena full of lava, the boss is too heavy to cross the bridges over it.
Transformers: War for Cybertron has this twice: Omega Supreme fights you near energon tanks you can taint with dark energon to make him vulnerable. Trypticon fights you near energon tank conveyor belts, which can be made to detonate and are the sole things capable of damaging his shoulder cannons.
James Bond: Everything or Nothing: In the boss fight after "An Old Friend," Jaws can be defeated by electrical bolts to his metal teeth. He shows Bond where the bolts are by pulling the four conduit pillars from the wall. Despite this, Bond still has to be careful: 1) those pillars hurt, 2) the electricity is a One-Hit Kill, and 3) Jaws' attacks deal at least 3/4 damage.
In an episode of Superman: The Animated Series, the Joker tries to kill Superman with a big hunk of kryptonite. He chooses to do this in a room filled with canisters of Hollywood Acid, which Batman uses to dissolve the kryptonite.