GrE writes, "There are games out there that just aren't possible to finish or "beat." There are games out there that you never WANT to finish. Then there are the games that you play over and over again, never getting any further than the play before, and for no reason.
First, games like Tetris, Lumines, and puzzle games of the like aren't really games that you finish. These are games of dexterity and skill more than completion. You play these games to show your friends (and enemies) that you are better than them and can stay literally in the game longer than they can. It's hard to count out games like these simply because pretty much everyone and their Grandmother has played at least one session of Tetris at some point. However given their style of play these are more for one-upsmanship than anything else..."