I think this depends somewhat what you mean by "competition." Do you mean something where you enter and are given some task (test, project, etc) to complete in a timed setting, or something where you submit information about yourself (personal statement, activities, publications, etc) and get awarded some prize?
If the latter, there are tons of them. So many that I have to assume this isn't what you are interested in. For the former, there are plenty but they're often quite difficult, specific to particular degrees/disciplines/societies, and often poorly advertised. For example, there is the
Annual Putnam Competition, or perhaps the
University Physics Competition, or one of various business e
levator pitch competitions. There are many essay competitions, short story competitions, foreign language exam competitions, etc etc. It's all about your interests, and finding socitieies/funding organizations that are interested.
I can't really recall a research competition besides stuff like travel grants or young-investigator awards. These are specific to the research organization in most cases but there exist others that are more broad. Perhaps you could explain why you want to do this to give us a better idea of what you're looking for.