I'm a completionist. X-Box Live Achievements are the worst thing that's ever happened to my social life, and I rarely put down a game completely until I have every achievement for it.
The exception, however, are multiplayer achievements. While I'm a HUGE fan of ones like Gears of War (the first one) and the "kill 100 people with the sniper rifle" or Chromehounds and the "Play on every map" achievement, some achievements seem like they're almost meant to ruin my spirits, such as Ghost Recon: Advanced Warfighter's "Be #1 on the leaderboard".
So to sum it up: A game is "completed" for me when I get all the achievements (X-Box Live or Steam) unless there are unrealistic multiplayer achievements.