I suspect that everyone likes to design and play the occasional odd character. Of course your definition of 'odd' may vary from mine. Personally I still want to play in an Atomic Horror game where my character has the ability to summon musical sequences - he can swing his guitar off his hip and give everyone the ability to dance in a musical number of that wacky rock and roll. I'm not sure what the in-game benefit would be (it doesn't even have the get more done during time compression montage effect that a Disney Princess' 'summon musical number' power does(, but I just think it's cool in the genre.
Let see... in one fantasy game in which I was to be an intermittent player my PC was actually a personality inside an amulet that would possess whoever put it on for short periods. Whenever I wasn't there the body became a combat ineffective commoner who the GM would not feel obliged to role play or make combat decisions for. The other cool part was that anyone I explained this too took POW damage and potential insanity, which meant that I could be all strange and mysterious and say "don't ask questions you don't want the answers to" and then when someone insisted, Player Character style that I absolutely tell them I could say to the GM "I tell him" and he'd roll for damage and describe the PC collapsing into a ball of blubbering temporary insanity. I don't know why I found that ability to whack at someone who insisted on trying to remove my PCs aura of mystery so enjoyable, but I did.
There was also my character in Tom's Bureau 13 game, a Banshee who felt really bad about past actions but had attended a twelve-step program and internalized that her wail was a prediction and not something she was causing and had to feel responsible for. She also dressed like a male dandy and each session I would slowly find a use for each handkerchief, pocket watch chain, silk tie and so on until at the end she was inevitably in a state of deshabille.
Those are a couple of mine - what're yours?