The first champ I ever bought was Orianna, and she remains to this day to be one of my favourite champions, due to her unique playstyle. She doesn't have the same burst of a typical AP carry, but I feel makes up for that with the utility she brings to the table.
I've also recently taken a liking to Rumble - he lets you play a very aggressive solo-top, and rarely do I have trouble shutting down my lane opponent(s). His DPS is crazy, and being manaless means that harass is basically free, and you gotta manage your heat to be most effective.
That said, I think I'm just attracted to champions with unique or unconventional playstyles :P