Thanks very much all for the helpful replies! The clock one is especially interesting. I wonder how fast he can make the clock hands move? It'd seem very interesting to me if it only took a bit to move from one random hour to the other.
After knowing servos are commonly 180, I set out seeing if there were any that were more towards 360. I think I found one, so I'm going to go with that and see how it turns out!
http://www.acroname.com/robotics/parts/R298-1T-SERVO.html
If that doesn't work out, I have a box of various stepper motors I'll get to play with.
Thanks,
-Jon