All the suggestions are great, but if it doesn't have to focus specifically on robotics, but science in general, Science Olympiad is an excellent competition. It consists of roughly 20 different events. These events cover many different fields of science (it's been a while for me so I don't know what the current ones are). Some involve building something and bringing it to the competition, and others you take a test at a competition to demonstrate your knowledge.
For more information about it, check out their website:
http://www.soinc.org/