some things I've tried or plan on trying:
-melee assassins: those single element skills are actually usefull. I'm building a 'blades of ice' assassin right now and I'm having great fun.
-maul werebear: just make a big fat tank with a slow melee attack. It's perfect for standing in front of monsters while some sorcs throw around meteors. Not a cookie-cutter, but with godd equipment can be quite strong....
-Golemlord (or whatever you want to name him): just waste those skillpoints in a very fat golem! and get some curses (1 point in all is fine) He can solo hell if you are VERY patient.
-freezing arrow or immolition arrow amazon: not very original, but if you never tried one before it can be fun.
-frog barbarian: kind of a singer who leaps around all the time, or you could get a strong leap attack, or whatever; leaping is FUN!
-sacrificial pala: not really a fullblown build but if you twink him with stuff like angelic set, twitchthroe, civerbs cudgel etc. you can try to defeat the ancients before you're level 20. It can be done! (I did it in 1.09 hardcore)
I'm sure there are many more fun/wacky characters out there, just try what you feel like, if it doesn't work out, no harm done, there's only a few hours of your life wasted, but if you're playing diablo2 they would be wasted anyway ;-)