House of the Dead: Chariot, the first boss, has only a tiny point in his breastplate where it takes damage. Pump enough rounds into the nick and he flexes his muscles, literally exploding out of his armor. Then you effortlesslyblast the flesh off his bones.
Targitzan. He's a giant totem composed by a head and four rotating pieces. As you shoot at the (literal) targets, the boss's height decreases.
Old King Coal, a monster made of coal loses body parts as you hurt him, first one arm, which he dismisses casually, then his other arm, which he's perturbed by, but otherwise unfazed, then the entire upper half of his body, whereupon he suddenly starts wondering if you'd like to sit down and talk about this.
Mingy Jongo loses a piece of his disguise with each hit you land on him, revealing his "cyboticness".
The last boss of Disgaea 2 has a number of shields/masks on him that disappear once he takes a sufficient amount of damage and with them, his resistance to elemental attacks.
Donkey Kong 64: Similar to the Banjo-Tooie example, also made by Rare, this game has King Kut-Out (a cardboard cut-out of the Big Bad, King K. Rool) loses first one arm, then the other, then his head, whereupon he dies. Given that the Kongs were firing themselves out of cannons at him, this is, perhaps, understandable.
Mangoruby in Donkey Kong Country Returns. The electrified caterpillar's body will explode into parts as it's hit by Donkey and Diddy, but this makes it faster and more vicious as a result.
Kirby: In Kirby Super Star, many of the mechanical bosses Kirby must oppose have extra guns that can be blown away.
Ocarina of Time: The boss inside Lord Jabu-Jabu requires the player to kill the jellyfish serving as his armor. Also, Stalchildren would lose their heads if you slashed them with your sword in a certain way. Being skeletons, this did nothing to stop them from attacking you.
Iron Knuckles from both Ocarina of Time and Majora's Mask drop off pieces of armor when you inflict a certain amount of damage to them. Better be on your toes when that happens, because it makes them much, much faster (to a lesser extent in Ocarina of Time, though).
Dark Nuts in Wind Waker become faster as they lose pieces of their armor and if Link manages to knock the sword out of their hand, they suddenly switch to a hand-to-hand combat style where they are actually more dangerous than when they had the sword.
The Legend of Zelda: Twilight Princess: Toward the end of the game, Link must wear down his heavily armored foes called Darknuts, by removing their armor piece by piece. Once it's removed, they toss their massive weapon and draw a longsword, and become capable of doing combos.
One boss is an armored dragon. Link must drag it down to the ground with the weight of his iron boots to crack the armor. After two incidents of this, the dragon will burst out of it.
Star Fox: In the original and N64 versions, the first boss is an aircraft-carrier of sorts that will lose pieces of its ship as you damage it.
The original did this quite a lot. Generally, at low health, bosses would lose some of their parts and go berserk. The Dancing Insector loses its legs, the Rock Crusher loses the giant middle part, Plasma Hydra loses its arms and gets a tongue, the Spinning Core loses its cover, and a whole part of the Great Commander is destroyed.
World of Warcraft: Many of the Earth Elemental enemies will begin to lose actual body mass as you damage them. Often it does weaken them, though sometimes it simply leads to smaller copies of the elemental attacking you as well.
The final boss of the Tournament of Champions five man dungeon, the Black Knight, goes through three incarnations as you battle him. First he is an armored Death Knight, then after you beat him down, he comes back as a skeleton. Finally he's a malevolent ghost. "My rotting flesh was just getting in the way."
It should be noted that, as both his name and the name of one of the player achievements associated with the fight would suggest, this boss is an obvious Shout-Out to the infamous Monty Python Black Knight mentioned under the film tab above.
The Legend of Spyro: Dawn of the Dragon has the Golem, which deliberately breaks off a body part to free itself from entrapment and later replaces it by assimilating nearby wreckage. Then its big brother the Destroyer gets about a third of its total body mass broken off in the course of the level where you fight it, including having its heart blown up, and keeps on going after the level ends, playing the trope very literally.
It also happens in A New Beginning with the Ice King. He has three health bars. With each depleted health bar he loses more and more of his armor, and then his skin, so that by the time you actually defeat him he's wearing nothing but torn rags and his limbs are reduced to exposed bone. I suppose that's the literal case of Only a Flesh Wound.
Clean Asia: The boss in China stage comes with four tentacles that spray bullets. Destroying them all makes the boss unleash an even bigger spray of bullets.
Dynamite Headdy has The Wooden Dresser, a giant wooden figure model that can't be damaged until you knock all of the clothes off of it first. A few seconds later, it will summon another costume. Then there's a semi-example in Baby Face, a giant mechanical Baby's Face that sheds faces for progressively older (and thinner, and tougher) ones as the battle goes on. If any of this sounds weird it's because it is.
The DS remake of Final Fantasy IV has the Octomammoth treat its first six tentacles this way.
Final Fantasy VII has Mighty Guards in the Shinra HQ. Mooks met early in the game, these are highly armoured and red. Once you beat them enough, the armour falls off, revealing a slender, greyish, and faster, but weaker, mook.
Every enemy in Final Fantasy Mystic Quest works like this. The mooks, the bosses, the final boss... Speaking of which, the Final Boss is the only one that changes attack patterns when you beat him up enough for him to change graphics.
Castlevania II: Belmont's Revenge features the Iron Doll. At first, it's a demon in a huge suit of amour that lumbers around slowly and swings its sword at you. After half his life is gone, he sheds his armour and gains a LOT of speed, leaping all over the place and shooting laser blasts from his sword.
Castlevania: Bloodlines has the Golem as the boss of Stage 2. You have to whip away at his segments, then whip his head which proceeds to blow up, revealing his core.
N. Gin in Crash 2 is a good example. He starts with two lasers in his mech's arms. You destroy those, and he reveals that his shoulders contain rockets. Destroying those make him use a cannon he had built into the mech's chest. Destroying that then blows his device up.
He does it even moreso in the next game. After defeating the same five parts, he goes into orbit, ejects some more parts, and enters a space station with even more weapons to destroy.
Destroying all seven of the miniature Robotnik Balloons in the Metropolis Zone leaves Robotnik to reveal his craft also has a laser to attack you with.
Every hit to the final boss of Sonic the Hedgehog CD will cause him to lose one of his four spinning arms, also changing his attack pattern entirely.
The main boss of the Ice Cap Zone, starts with a platform that he raises up to goad you into attacking while he tries to freeze you. It gets broken off after six hits, leaving him to just float around spewing the freezing gas.
The second boss of act two of the Launch Base Zone will lose both his laser cannons leaving you to just try to hit him as he flies up and down.
The sub-boss of Lava Reef has two bullet firing tentacles that can be destroyed. It doesn't make any difference to the fight if you do or don't (save for making one less obstacle to avoid), as the boss ends once you've hit the giant hand six times either way.
The Final Boss of the Death Egg Zone in S3&K will start attacking you with just his giant fingers. Once you smash those off one by one, he reveals he's turned the Master Emerald into a powerful laser cannon.
The boss of Panic Puppet Zone will simply lower down to another floor if you destroy the two weapons on that floor, until you've destroyed six separate weapons.
Every nonhuman enemy (except the final boss, who possesses instant regeneration) in Valkyrie Profile 2 Silmeria is subject to this: all major body parts have their own hidden HP count and are severed when it is depleted, allowing the party to end up fighting huge dragons with only a single limb left if they so desire. This also ties in with Randomly Drops, in that severed enemy parts have a set percentage of becoming items, but thankfully never dropping below a 15% chance (due to the difficulty of actually chopping off the right enemy part without killing the enemy outright, as you can't specifically target the body parts themselves, and exactly what you end up hitting is determined by what attacks you use and the enemy's position.)
Done rather brutally in God of War 3, where during the battle with Hades, Kratos must rip whole chunks of flesh off his body. Then you have to "kill" the chunks before they slide across the floor and reattach themselves to Hades' body.
Also done with Herakles, who speeds up as he gets his armor knocked off.
Herc also kinda paraphrases the trope name once his last piece of armor is gone: "ARMOR IS FOR WEAKLINGS!"
The original game has the giant armored Minotaur fought in the temple. You knock off pieces of its armor until the beast itself finally becomes vulnerable to damage.
Gets done to a ridiculous extent in the Monster Hunter series. Almost every boss has "breakable" parts, which either scar or are removed outright from the body after enough damage is on a centralized area. It gets ridiculous when you break a monster's beak, back, claws, and sever its tail, yet it still continues to attack without showing an ounce of pain. There is also a monster, the Barroth, which can have part of its skull severed and it still fights at full force.
God Eater Burst, which is... loosely inspired by Monster Hunter, has the same thing happen with most of its enemies (aside from the smaller ones) - they have three breakable parts, but breaking them only grants you bonus materials at the end of the mission. For example, the Borg Camlann is a giant scorpion-styled enemy with a lance-like needle on its tail - break the needle, and about a third of its length busts off, but the Borg Camlann can still attack with it just fine and it has the same hitbox (which can result in it sometimes jamming nothing into the ground with a powerful attack and struggling to pull it out). A bit disconcerting in a game that has otherwise excellent mapping of hitboxes to models.
Bayonetta has the Cardinal Virtues, a quartet of huge bosses that you face throughout the game. As the fight progresses, you rip off parts of their body using Prehensile Hair, which at best makes them turn red. However, at the end of each fight, despite having at least 50% of their body gone, they give their last words as though they weren't in some sort of excruciating agony.
Applies to normal enemies as well, which undergo a pretty severe Glamor Failure when they're near death, as their muscle tendons become exposed and gooey liquids drip from what's left of their skin.
Angel-like enemies that seem much less majestic once you've hacked at them a little is yet another thing Bayonetta carried over from Devil May Cry - in the third game, Dante faced off against four-winged angels who would use two wings to fly and two to cover their bodies with an invulnerable shield, and only be hittable when they dashed. After a couple hits, though, wings fall off, revealing snarling demonic faces on their torsos.
Portal 2 inverts this. You defeat the final boss by sticking different cores back on.
Plants vs. Zombies has zombies with armor you must destroy (or remove some other way, like with a magnet shroom) before being able to defeat them. And their arms fall off when they're really near death, finishing off with the fact that the heads fall off when they finally die.
Ace Combat: Joint Assault has Sulejmani's Varcolac. At first, it has a rear-facing point defence machine gun that destroys any missiles coming from the six. If you manage to damage it enough, though, it loses the PD gun and gains the ability to do ridiculous missile-dodging manoeuvres you could never replicate.
The first boss of Conkers Bad Fur Day, Haybot, loses parts of its body as Conker and Franky continue pressing the red button behind its body. When Haybot is complete, it attacks by squashing the characters with both hands. When one of those hands is gone, it attacks by seizing them and then throwing them away. With both hands gone, it squashed again the characters, but with its own metallic base. With the rest of the body gone, the boss is simply defeated.
Quadraxis from Metroid Prime 2. This is the reason why it's a Marathon Boss, as dismembering it part-by-part takes a very long time. It's only truly defeated when its head module is completely destroyed.
The rock creature Thaardus from the first Metroid Prime is defeated by scanning its body for the current vital spot, and then blowing up that segment of its body in order to reveal its vulnerable core. After each turn, the weak spot moves to a new segment that has not been destroyed yet. Thaardus seems entirely unaffected by its missing parts, which makes sense given that the creature itself consists only of the core. The Omega Pirate is another giant beast with different armored segments on its body, though in this case destroying the armor and its weapons actually forces it to cloak and flee to regenerate, at which point it can be properly harmed and killed.
In the third game of the series, the Metroid Hatcher has four tentacles that must be individually torn off. Once they are gone, the creature dies, but the loss of any individual tentacle does not hinder it.
Lar, the final boss of Chariot - Adventure through the sky (the Shoot 'em Up part of the arcade collection Three Wonders) loses its entire body when you hit him enough, leaving only his floating head/mask.
This happens to Drilldigger in 'Mario & Luigi: Dream Team after every hit, with bits of machinery and cogs flying off when it gets smashed into the background or just plain countered in general. Of course, this makes no real difference to how it looks or fights, so it makes you wonder where exactly those parts fell from anyway.
Your own zombie minions in Undead Knights, who can lose various body parts but still perform the exact same attacks - send a bunch to mob a guy, and your headless zombies will still act like they're biting and chewing on him just the same as your intact zombies.
Dwarf Fortress: Normally losing An Arm and a Leg will seriously hinder your average dwarf or other creature. Anything with a surplus of limbs, giant cave spiders in particular, will likely shrug such attacks off. And since combat AI more or less picks which body part to aim for at random, anything with an abundance of body parts to target will give your dwarves hell.
The Matrix Path Of Neo has this with the final, final version of Smith, when both his arms are gone he just gets even angrier.