I've had only three characters that were interesting at all, and none of which lasted very long, so I'll just talk about the little snippets I had with all three.
There was Something-or-other, a rogue from a small town that had literally no magical resources at all save for a few books that made mention of it. From these resources, Something-or-other concluded that all "mages" were simply charlatans and alchemists, so deciding he still wished to command their respect, he learned how to make alchemical items, he dressed in silly robes, and he trained a hawk to perch on his arm. It was fun while it lasted, but the only adventure I got to play him in was basically just a big battle with a couple of ogres (our DM was a big War gamer)
There was also Arog, the half-orc fighter wannabe paladin (I hve a lot of wannabes...) He desperately wanted to be a Paladin, and though he trained as a paladin and had a good heart and a sense of valor, he still couldn't quite get over his chaotic bent. He was fun to play until the DM decided there was no point to a game with two players.
Finally, there was Avellios Moonwhisper, an elven swashbuckler. He had spent much of his life as a pirate on the seas, though he was careful not to do anything "immoral." His father died long back in an attack on their village by some extra-planar entity. This gave the elf a reason to adventure, as well as a strong hatred of evil. The best moment with him is probably when he tried (in vain) to convince the party NOT to try and break through the floor to see if there was a lower level.
Monk: Okay, I'll try and punch through the floor!
DM: You say ouch.