Cheep-Cheeps in the Super Mario Bros. franchise. Unlike virtually every other enemy seen in the franchise, they respawn constantly and their movement makes them difficult to avoid by simply running fast. And like every other enemy, they drain your life or kill you with any touch that isn't a Goomba Stomp.
Frankenstein's Monster for the Atari, as seen here, 9 minutes and 40 seconds in, "Look at all these goddamn bats!"
A game called Pharaoh's Curse for the Commodore 64 has a very pesky bat (or is it a bird?). It doesn't kill you at all. No, what it does is swoop down, pick you up, and carry you to a random spot, forcing you to drop any key you were carrying, leaving yourself unable to open the locked door you were planning to. Of course, you can only shoot sideways, so you would be often helpless to avoid it when it swooped down from above at high speed.
Some segments of Viewtiful Joe feature annoying swarms of infinite robotic bats that stick to you, stunning you, and explode unless you button mash them away. They make you solve puzzles slower, and allow other enemies to hit you, but can be fended off with a Hurricane Kick.
There's a certain type of robot enemy in Wario Land 4, found only in Hard mode and above which pretty much covers this. It's what appears to be some kind of metal bird creature, flies in the air right near platforming sections and charges at Wario when he gets near. Not hard to defeat (one hit kills it), but considering that it's found in narrow spaces, doesn't give you anything for killing it, and respawns when you re enter the area, even though every other non transformation enemy doesn't come back unless you win/lose the level, it's freaking annoying. Plus, Pinball Zone (yes, that Pinball Zone that named the trope) is annoying enough without them.
Trine has literal bats that: Always appear at the worst moment (like when you're trying to avoid falling into a Bottomless Pit), always travel in packs, are hard to hit (only one of the characters has an air attack) and deplete your Health Meter very rapidly.
The Swoopys in Donkey Kong Country 3. Little woodpeckers that you think are a Unique Enemy... until you reach the Lost World, where they get a level all to themselves. They fly across the stage in never-ending queues; killing one can make a gap in the line, but there will always be more. Most of the level you have to fly past them as Squawks the parrot, with squirrelly controls and a hard-to-aim arcing projectile attack. To reach the bonus levels, you have to slip through the knotholes they use, going with them once, going against them another. And they're in the bonus levels too, resulting in failure if one hits you!
Zingers become this in Donkey Kong 64 where they dive bomb the player and drop grenades. They can be shot out of the sky, but they just respawn shortly. Goddamned Bees...
Donkey Kong 64 also gives us Purple Klaptraps. These things will chase you down if they see you, and are immune to all standard attacks. There are two ways to kill them: orange bombs and your musical instrument attack. The former is horribly inaccurate, and can actually hurt you if you aim it wrong. The latter is the rough equivalent of using a nuke to kill a squirrel. The fact that these things don't drop health when you kill them does not help.
the crabs in the beach levels, due to taking too many steps to defeat for how common they are: first, you must jump on them, making them lift their pincers and makes them Spinies. Then, you must hit the ground to flip them over (if you do this before jumping on them, they use their pincers to stop themselves from flipping over), then you can jump on them and finish them off.
Crystal Caves has the birds (in underground caverns??) who fly high up, beyond the reach of your horizontally-shooting gun, and bombard you with hard-to-dodge eggs. And the eggs remain on the ground for a while after they fall and crack, and can harm you when you step on them. And sometimes they hatch into another, smaller bird when on the way down.
The Snuggle(?) Bunnies in Nicktoons: Attack Of The Toybots. They spawn in about groups of two or more, are small and difficult to hit, and once they see you they'll try and cling to you, rendering you unable to attack. If you don't managed to shake them off in time they'll explode, flinging you merrily into the air, and often off a platform.
Bug!! With six worlds, each world is bound to have one:
Insectia: Bees. NOT THE BEES! The first Airborne Mook that Bug will encounter, they will try to swoop in on Bug at an awkward angle, and Bug will have to time his jump exactly in order to beat them. Some of them even come in from nowhere, just when you thought it was safe!
Reptilia: Army ants. As in, ants that drop via parachutes, wear army hats, and fire bouncy grenades from their tails. The main problem is that they tend to be positioned at the top of an incline, makign it hard to dodge the grenades.
Splot!: Frogs, frogs, and more frogs. These things jump quite quickly, take three hits to die, and their tongue attack has good range. There are also literal lightning bugs, another Airborne Mook that attacks from an awkward angle to hit and fires out a stream of lightning at the floor.
Quaria: Sea worms move quite fast (especially when they start rolling when you get near) and take five hits to die. Fire-breathing moray eels are also very annoying because Bug starts out so far from them and will have to get closer while avoiding their fireballs.
Burrbs: Snow fleas. Some just jumped out of the floor here and there. Some jumped out of the floor and threw snow at Bug. Some stayed at one point rolling snowballs at Bug until Bug got near, at which they ran away. And the worst kind were those who assaulted Bug with snowballs from a distance!
Arachnia: Green Flies that vomited a pool of acid on the ground. They came in large numbers from offscreen. Also, there were spiders who rolled bowling balls at Bug. Not too bad on level ground, but they usually did it from the top of a slope...
Spyro: A Hero's Tail has some literal Goddamned Bats in the Crocovile Swamp level - rest assured that if you enter a cave or tunnel at any time in the level, several bats will fly out. They deal damage just by touching you, and unless you're psychic and know they're coming, you'll be too late to kill them every time as they just fly away after taking half your health.
Taz-Mania for the Sega Genesis has the goddamned Bushrats. They're almost impossible to jump on due to their size, they throw spears at you, they can cause you to accidentally spin away health powerups in the process of killing them, and they can also cause you to spin off a platform and into oblivion.
Also, the actual bats in the ruins levels. They can camouflage themselves against the dark background to make themselves harder to see.
Mario vs. Donkey Kong has certain levels that feature an invulnerable bird that flies back and forth and drops eggs. If the bird or one of its eggs touches Mario, he loses a life, forcing you to restart the level from the beginning, and unlike other Mario games, the bird can not be defeated by jumping on it. The levels it appears in are considerably more frustrating than they reasonably should be.
In Atlantis No Nazo, Batabats fly so high the player can't even touch them and throw down triangle-shaped droppings. They're the first enemy encountered in the game.
The ice bats in Super Mario Galaxy. Mostly appearing in Freezeflame Galaxy, they are a pain to kill, especially in the secret mission, in that they can only be killed as Fire Mario, which can often be hard to aim. Not to mention the fact that they freeze you when they touch you, and if they hit you in mid-air, it's all over. Not to mention the fact that they can follow you, so running isn't an option, and they swoop with deadly accuracy.
Any ranged enemy can qualify. Jumping for jeweled skull? I don't think so, enjoy your damage stun which will send you plummeting into spikes.
Seagulls. Attack you from most various of places(usually in places where your jump accuracy needs to be perfect or near-perfect), can see you with invinsibility powerup and in 9 cases out of 10 give you nothing in return. "Bloody birds" indeed.
Crabs of bomb-throwing variety. Not only they pop out of spawns at the worst possible moment, their bombs have quite a range. While not damaging, they can still distract you enough to lose bonus points/more hitpoints/even a life. Finally, when hiding in their shells they're invulnerable to anything short of a Magic Claw.
Fishes serve you as a seagulls of underwater variety. This should be enough for your imagination.
And finally, Chameleons which could count as Demonic Spiders at first. They make themselves invinsible which makes them immune to all kinds of damage and they have a dickish tendency to re-appear just on you, causing Collision Damage, followed-up by an attack which will cause you to lose another 10 hitpoints.
Crescent Pale Mist crosses this with Demonic Spiders as enemies of the game can come in great numbers, can hit hard, and put Yunou in a series of nearly inescapable attacks. One particular area of Chapter 4 is littered with many, many Demonik Blades and a Shuvalzard among the bunch.
And another area in Chapter 4 houses a boss enemy known as Gilgamesh (a golden Magic Armor with powerful attacks), Dark Armors, and Gargoyles. The worst part: You are trapped until the enemies are killed.
Super Mario World 2: Yoshi's Island has Grinders, monkeys that plague World 3 and are a real pest. They're fast, take multiple hits and have several methods of annoying you; such as grabbing Baby Mario right off Yoshi's back and running off, spitting seeds at you and dropping bombs, which make holes in the level, making it easy for you to fall to your death. You will want to go to the nearest jungle and exterminate every monkey you can find by the time you're done with World 3.
Adventure: In certain difficulty levels, a bat would fly throughout the game stealing whatever item the player was carrying and exchanging it for another, or sometimes just outright taking it. This could prove especially annoying as the bat would sometimes steal the player's sword, and replace them with, oh, say, the DRAGONS.
Cave Run 3D has these annoying groups of purple bats which, while they don't actually harm you, crowd around you and slow you down just enough for your character to have a higher chance of missing the next jump and going splat.
In level three of 'Sparkster'' for the SNES, on your way to the robot caterpillar boss, you have to make it past robot bats (presumably the same ones from the original game on the Sega Genesis/Mega Drive) without getting hit when being launched from one robot ostrich to the one you battle the robot caterpillar with. As if that weren't enough, just before you fight the robot caterpillar, you have to avoid bomb-throwing robot monkeys (destroying them is optional). Both the robot bats and the robot monkeys are guaranteed to take away at least half a heart. When you do finally make it to the robot caterpillar, you're lucky if you have enough health left to survive while fighting it.
Roguelite platformer Downwell has literal bats that cling to the walls until you're below them, then flutter slowly towards you. The problem is that this is a game where you go down and your weapons also only fire downwards. By themselves, they're fairly easy to dispatch by waiting and jumping on them, but mix in other enemies and they become a nightmare.