IEEE, Institute of Electrical and Electronics Engineers, hosts a robotics competition in region 5. I am not sure about the other regions.....This year it is an autonomous maze solving robot. It is much more.....programming intensive, for lack of a better word, than most of FRC in my opinion. No external sensors are allowed. You are free to design your robot however you'd like, but it really comes down to how good of an algorithm you write analyse the maze and solve it.
Link for info
Some other things that colleges have, that others mentioned, are design teams. I don't know much about them, but I know there is a solar car competition, human-powered vehicle, and Mars Rover.
If you don't find anything that you find interesting, look into getting into research/working at a lab at for university.