We had a character in our campaign who was going to be a thieving con artist. He decided to try and con a noble, he wasn't there. So he tried to steal some stuff of of his desk, and there was nothing of value.
Then came in the dog. He tried to kidnap it and hold it for ransom. The dog proceeded to change over the next three sessions into a lambrodoodle. Half lab, half poodle, half sheep, and half marijuana plant. It proceeded to survive every piece of crap the GM threw at us and become the running joke.

And it was all caused because a person made fun of the GM's mispronunciation of the word Labradoodle.

While i can't recall any such cases from campaigns i've played in, i frequently find myself doing stuff like that in games i run myself.

I'll give 2, one where I was the GM, and one where someone else was.

When I was the GM, had villains black out the city, only one player for some reason decided to go to the power plant, but thanks to being insanely well armed, and good rolls, he took out all the major villains present, but scared a lowly henchman into running. He tried to kill the henchman, and somehow, roll after roll, he'd miss and the Henchman would roll 19s and 20s, so finally he got away, but one of the guys called him Lucky.

the Lucky Henchman was brought back a few sessions later, and somehow, now facing a full team of heroes, he still seemed untouchable, dodging, saving, and having them miss him.

So Lucky was born. I would bring him back, time and again, campaign after campaign, and yet, even when one Hero tried a nuclear option (seriously, guy hated Lucky so badly he essentially bombed a huge shopping center to take Lucky out...) Lucky survived and thrived.

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Okay, the other one, after I'd played 'Cows with Guns' and 'I want a Hippopotamus for Christmas', the GM kidded about a character being a Hippo with Guns... sufficed to say, I helped a new player make a Mutant Hippo with Guns and a Force Field Power... Drove the GM nuts with the Midget Hippo Mutant Running around with dual M16 action.

Wasn't there a race in Spelljammer that were Hippos with Guns?

I was GMing a game and I read a comment on DarthsandDroids about adding in random magical objects like a stick that stood straight up or a rope that always untied itself and wanted to try it with my campaign. So, I decided to give a brick that floated with no other bonuses, just to see what happened.

One of the players got it into his head that this brick was the best thing since sliced bread and tried to use it to solve every problem, from trying to use it as a mobile cannon platform to putting it under the feet of a large monster. Drove me and the rest of the party crazy.

In the same campaign where the evil cleric called on his deity and was cursed and sent on a quest for his trouble, the other cleric in the party decided that he didn't like the leader. It was good role-playing, as the clerics served different divinities and were of different alignments, but he decided to do something about it.

He convinced me to let him contact the local thieves guild and recruit some guides for the party. He did, but he also paid them extra to backstab the leader when it the time was right. The rebel cleric also recruited a bunch of horse barbarians to ambush the party on the trail. The idea was that the guides would assassinate the leader during the ambush. He forgot one thing; he never actually role-played telling the guides about the ambush.

So when it happened, they hung back, until the horse barbarians were starting to lose, then launched their attack. I justified it by saying that they didn't want to share credit with anyone else. They didn't kill the leader, but they escaped to harass the party all the way to the end of the campaign, even following the party into Hell. That was something I would never have thought of myself.

In an Exalted game I ran, the party was given a (very railroad-y) direction to where they were supposed to >start< the adventure I had planned...

Instead, they decided to sort of sight-see/investigate before they got there, and so demanded to go to the planning bureau of the city. Trying to roll with it, without putting too much effort in, I informed them that they needed to visit the office of a minor spirit of paperwork that worked there named Red Smelt.

None of them caught the reference.

On later reflection, I'm not sure they'd have noticed if I'd named the guy something like "Crimson Herring"... ^^;

Still, it was a fun session, and the annoying-as-hell, would-be-but-very-bad thief-type actually gave the group some good laughs, going through all sorts of fumbles and pratfalls, and so did his character... ^_______^

Ever heard of Raggamoffyns?

They're sentient attack clothes capable of possessing people.

First time I ever saw them, we'd been following the rogue through the dungeon on a final Loot sweep.

Went well, got an amazing sword out of it with matching armor and shield.

Anyway, so the rogue opens up a wardrobe and gets pulled in and strangled to death, so we bail the hell out screaming about mimics.

A badly dressed corpse ended up following us around and showing up in the strangest of places, Like Hell.

So we were playing in a city we just arrived in via stolen slaver ship. I was a half giant (Fire Giant Dad), and we arrived at the *2nd* dingiest/cheapest tavern in the lower end district. we were very insistent on finding the not-quite-worst bar to set up as home-base. anyway the tavernkeep was a dwarf, and I asked him for a drink. GM said it tasted like Dirt. so we just assumed it was Dirt Beer. which my character decided to take a shine to. "acquired taste" he called it. so the GM came up with this whole backstory for this dwarf who brewed his own ancestral recipe for stout...using fermented Dirt. Literal Dirt Beer.

One player come up with a backstory for his character that involved a Tribe of Martials-artist-shapeshifting-fire-ape, who used giant leafs to surf on the air.
APE.WITH FIRE. AND MARTIAL ART.
I accepted this backstory, and of course i made thoses thing appear during the campaign.

In a Shadowrun game I played in, one of the PC's was a snake mage (not a mage with snake themed magic, but a snake with magic). During the first fight of the snake mage's first run, when it was his turn and he asked us what he should do I told him that one of the goons attacking us made a racist comment about him, and that he had a monocle. The GM liked this, and said that it was now actually a thing

The snake mage's player decided that he really wanted that monocle, so he spent the entire encounter slowly killing the goon with a spell that turned him into slime, just to make sure the monocle wasn't damaged.

And that is how the Dapper Snake was born.