A gelded cleric. Don't know why the player wanted her cleric to be missing certain "equipment" (she'd played intact male characters before without a problem), but for some odd reason she wanted this cleric to be a castrato singer (with the NWP for singing, this was in 2e).
A "masked mage". This was made with 2e Players Option: Spells & Magic. A PC (who has a habit of coming up with "cool" character concepts that get killed in their first session or two). Essentially, he took the flaw that each spell required a unique focus to cast, and he chose masks. So, in addition to memorizing out of his spellbook, for every spell he cast he had to put on a special mask, specific to that spell, and he had to carry all those masks around with him, and each new mask took time and money to create. This got real impractical real quick, as you could imagine shuffling through an array of face masks for each spell you want to cast would be, especially when the DM really enforced that penalty (encumbrance for the masks, having masks damaged or stolen, having to have masks on him at all times to cast). For this he got a few character points, which I think he spent on being able to buy weapon proficiencies as if he was a rogue, which meant he was proficient with a shortsword instead of a dagger. Woo, that's what you got for all that hassle.
Anthropomorphic Raven Bard (in 2e, no Savage Species or anything, not an anthropomorphic/furry game either, the same player as the "Masked Mage" insisted on playing a smart-mouthed anthropomorphic raven bard who wielded twin wakizashis, in a Forgotten Realms game with a novice GM. After much whining, the DM relented and came up with homebrew racial stats to reflect this concept, which lasted about a half-session before he pissed off a Chaotic Neutral PC Invoker with one of his "witty" sayings, and became a "Friendly Fire" casualty when he was in melee, then a huge rant and hissy fit about his character dying. I wasn't the GM there, but I was one of the players who had to see that go down.
Kobold psion. In a fairly mundane D&D game (the DM allowed any +0 ECL races and psionics, but he was the only PC to accept that offer). It got weirder when he started to roleplay the character as a character who had totally eschewed materail possessions. Now, this was before BoED came out so VoP wasn't an option, but this character was far from Exalted, he (in game) lied and stole, he just felt he had his psionic powers so having magical (or psionic) gear was beneath him and he didn't need wealth. He then gets angry because the characters is seriously underpowered, which he blames on Psionics being worthless, and refuses to believe it has anything to do with a lack of magic items. By the way, same player as the Raven and Masked Mage.
That same guy had yet another weird concept. As he is obsessed with ninja, but no GM would let him play an OA ninja or anything out of Complete Ninja's Handbook (again 2e era), he created a ninja/geisha character using Skills & Powers by modifying Thief, but to have the ability to take weapon specialization (and a few other changes that I dont recall). Instead of going with a sword, or even shuriken, this was a ninja who was specialized in chopsticks. Yes, chopsticks. There were rules for them in 1e OA, they did like a single point of damage, and he would go into battle poking enemies with chopsticks for a point of damage (with the 2e strength rules, no strength bonus to damage, and eventually the +2 for being specialized). Like the other characters, he got real whiny when this character wasn't as cool as he imagined it, so it only lasted a relatively short while.
Lastly, a psychotic gardener (in a d6 Star Wars campaign) who took dead stormtroopers and tried to hollow out their skulls and torsos to use as planters and watered his plants with the blood of his enemies. After a few DSP's, and a "What the heck?!?" when the character was played (it was pitched to me as a garderner who became a Rebel), that PC dropped out of my game. At least this was a different guy.
A moralist cleric in a 2e Al Qadim game (the faction that could be called "fundamentalist"), that was played like a stereotypical street preacher (complete with verbal quirks to emulate certain stereotypes best left unelaborated on this board), except with a weapon the player called "grim bopper", a truly huge hammer that the DM mentioned once years before he'd allow, and was called on it in this case (it did huge damage, but had a speed factor of like 15, so you could only drop it once a round, and spend the rest of the round picking it back up, and the fight might even be over before you could drop it on your foes even once).