Unconscious guards in the Metal Gear series are differentiated from dead guards by a little halo of stars merrily orbiting above their heads. As the guard gets closer and closer to waking up, one star at a time will cheerfully ascend heavenward and vanish. A guard with five stars won't be waking up for a while, but a guard with one star could open his eyes any time... (Weirdly enough, the MGS games are usually considered serious. Of course, it does fulfill a gameplay function.)
In The Legend of Zelda: Ocarina of Time, when sneaking around the Gerudo Fortress, Link can stun Gerudo guards with either his Hookshot or arrows. If arrows are used, then the Gerudo falls down, and has stars circling her head.
Technically, the ones in Gold/Silver are bird balloons.
In Pokémon Heartgold and Soulsilver, in the minigames, the birds are replaced by Psyduck.
In the Pokéwalker, the player's Pokémon experience this; when knocked out, in the journal, stars circle that Pokémon's head, Blissey for example. For large Pokémon, the stars that circle its head are not in the background, but above the background. Wild Pokémon do not experience this trope; when the HP hits zero, they run away.
Also in the Gen V games during the Poke Transfer minigame, circling stars appear if two Pokemon bump into each other by mistake.
Bubble Bobble (not in the NES [or Virtual Console] version): When Bub/Bob dies, after he spins out, he falls on his back and dizzily sees stars above his head while his eyes circle around (no spirals) before poofing away. It's also in the first Puzzle Bobble/Bust-A-Move'' game, in which the sprite animations are based on Bubble Bobble's.
Bubble Symphony: Bub and Bob still fall backwards upon death (even though they spin out differently; their original spinouts being taken by Coro and Kulu) and their eyes circle around but no stars are visible.
In Puzzle Bobble 4 / Bust-a-Move 4, when the player loses, Bub/Bob immediately gets dizzy, gains spiral eyes, twirls around, falls forward, and has birds circling above his head. Also the case for Bub in the Game Boy Color exclusiveBust-A-Move Millennium, and Bob in SPACE Puzzle Bobble. (Bob the unlockable character cries instead in BAMM.)
Bubble Bobble Plus (WiiWare): They spin out as fast as in the arcade original and end up lying on their back before they poof away. There are no hovering stars. According to this.
In Team Fortress 2, a player hit with a baseball will be temporarily stunned and treated to a third-person view of their character in a dazed trance, complete with "BONK!" written over their heads with stars circling it.
In the Wii Punch-Out!!, when an opponent is stunned, something related to their nationality/gimmick will circle their heads. For example, Piston Hondo has circling egg sushi.
Unsurprisingly, as it's basically a knockoff of the Punchout series, Glu's Super KO Fighter 2 uses the trope in the same fashion when a fighter is knocked down. The fighter 15 Cent, a parody of 50 Cent, sees circling dollar signs, for example. Your character sees simple stars. The trope actually factors into gameplay, as the fewer circling items above the character's head, the more likely he is to be able to get back up.
Happens in Monster Hunter to monsters if they get smacked in the head with a blunt weapon enough times, or if they get hit with a flash bomb. Can also happen to the hunters if they get hit with certain attacks, or receive too many hits within a short span of time (In which case the number of stars deonotes how much time is remaining in the stun).
Circling birdies denote being stunned in the Tales Series, which can happen to pretty much anyone who suffers enough abuse in a short period of time, or if they get smacked upside the head with a Pow Hammer.
Any enemy stunned by Commander Keen's Neural Stunner in Keen 4, 5, and 6 will see stars around its head — if you don't, that means the stunning is only temporary. Fortunately, this applies to Keen himself if he gets stunned by his own fire in 6.
The final frame of Mario/Jumpman's death animation in most versions of the original Donkey Kong includes a plain white halo. It was likely intended as this but limited by hardware.
In the first two Donkey Kong Country games, Diddy will get a ring of stars over his head if you lose a life while playing as him. In Donkey Kong Country Returns, enemies you stun with the peanut popgun/ground pound get an actual birdie circle.
Most of the Mario Party games feature this with birds if a character is stunned by something like dangerous areas or hazards on the game board.
In Super Smash Bros. Brawl, any character stunned by a broken shield or reflector, exploded Deku Nut, Mewtwo's Disable move, Weavile's False Swipe, Wario's Wario Waft, or Luigi's Negative Zone will be rendered immobile and a tweeting sound effect is heard. Button Mashing may get the player out of the "stunned" effect sooner.
In one cutscene of Sonic Unleashed, as Chip explains on how he lost his memory, when he said "And then nothing," he sees stars and tiny versions of himself circling his head (Wii version only, for they are replaced with normal stars in all other versions), as seen here.
Dr. Eggman did not get the same thing; in another cutscene, Eggman sees stars after a kid throws a rock at Eggman's head, as seen here.
Creature World also has Seeing Stars Head Bonk, and Seeing Birds Head Bonk, in which users see stars, or birds when they wear the hat. There is also a hat in which users have hearts in their eyes and hearts circling the user's head.
Bomberman games sometime feature this with birds or stars when players are stunned. In Bomberman Blast, if a player is stunned, the tweeting of birds can be heard from the Wii Remote Speaker.
MapleStory has this trope; players and enemies that are dazed get a halo of circling mushrooms. This happens after taking damage by a certain enemy and so on.
Toonstruck features circling stars at one point. And you need to take these as an item.
The Neverhood: While they're not shown, falling to the bottom of the drained lake for the first time will result in a cutscene of Klaymen being stunned, which is accompanied by typical "birdies" sounds.
In Epic Mickey Mickey sees Gremlins after getting slammed in the face while freeing one from a safe. The gremlin then flies over to him, takes a moment to glance between Mickey and the circling versions of himself, then comments on what a good likeness they are.
Power Instinct has this starting with the first game, but one of the stuns are flying hearts. The second game takes this to the next level with circling objects, such as people, sushi plates, chicks, and fish.
So you are playing Fighters History, and you fight an opponent. However, if you hit a weak spot multiple times and when a character is stunned, this game has circling objects that differ from each character, similar to Darkstalkers. Take Liu Feilin for example. When Feilin is stunned, she sees circling fairies. For another example, Liu Yungmie sees circling unicorns when she is stunned.
Happens often in the Raving Rabbids series after they get hit on the head with an object.
In Crusader of Centy, the first boss, the Big Bad Wolf, attacks by hitting himself on the head with a hammer, causing stars and birds to circle his head. He then uses these objects to damage you.
In Sheep, Dog 'n' Wolf, Ralph gets stars circling over his head when running into a wall. And in one instance, Daffy Duck gets them too.
In World of Warcraft, a character can get "dazed" if a monster hits him from behind, resulting in a 50% reduction to his movement speed for 3-4 seconds. While he's dazed, an animation of swirling stars appears over his head.