There is a game which does exactly this. It's called Chronotron, and you play a time-traveling robot who has to create multiple versions of himself via time travel in order to complete each stage. The loop must be closed for all iterations, though, which means that if a future version of himself interacts with a past version of himself in such a way that the past version is prevented from going back in time, you lose the stage and have to start over.

Pretty fun game, although since it's a Flash game it's relatively short.