Tivi, who is seven, just taught me and Jim a nifty game we hadn't seen before. Tivi called it “Sticks”; Wikipedia calls it “Chopsticks.”
Unfortunately, the Wikipedia article's attempt to explain the rules is awful, and very confusing, and I don't have time to improve it right now. The eHow page about the game is better, though still somewhat confusing.
The version Tivi taught us is a variant combining the “Splits” rule (where you can redistribute fingers between your two hands) and the “Leftovers” rule, in which going over 5 wraps around to n-5 (or n mod 5, if you prefer to think of it that way). I like the version he taught us better than the basic game.
Various pages suggest that it's a two-player game, but we played with three and it seemed to work okay. (But we only played once.)
I see that there's also an online version, allegedly with a computer opponent, but the computer doesn't seem to actually play.