Lets see. . . .
Nidalee, Zilean, Morde, Rammus, Malph, Yi, Nunu, Caitlynn, Udyr, Ashe. . . .
About that many, probably 1-3 more I forgot about.
I haven't actually played a game with each of them, but those are the characters I am comfortable with so far(in ranked).
What Arcadies said about supports/tanks/carries is good. You need to be able to pick something your team doesn't have in ranked if you need too, so it's important to be capable of filling any roll to balance your team.
If you're doing premade 3v3 or 5v5, then it's not so important, but it still helps to know multiple roles.