Any two player game will work, tic-tac-toe, fences, checkers, tetris. I particularly enjoyed a programming competition that involved a real time tank battle, like "combat" on the atari.

Any game that's grid based with a fixed number of states works well, either turn-based or real time.

You can do a double elimination bracket like a sports tournament, and can be done on a spreadsheet or whiteboard.

For more than two players at once, you could try snake, hearts, or poker.