It varies from game to game. I've been replaying through Batman: Arkham City over the past week or so (because I replaced my Xbox disc with the PC version), so originally, I was going to say that I finish most of my games when I beat the main story but don't consider a game with side quests finished until I've scrubbed all those pesky alerts off my map. But then I considered Pokemon, where the primary side quest is catching them all (or... is that actually the main quest?), yet I feel no drive to make the attempt because it's impossible. I suppose therein lies the difference, then; side quests only feel mandatory when they feel reasonably achievable (come off it, Minecraft, the amount of iron it would take to lay a kilometer's worth of rail could practically build a full-sized MechaGodzilla).
Achievements come into it in a very few instances, though. When playing Halo Combat Evolved: Anniversary, I had already played through the original game more times than I could count (even on Legendary), so 100%ing the achievements became my new goal for the remake. I also made a point in getting all the achievements for Arkham Asylum simply because I loved the game so much (although I would have felt compelled to get perfect scores on all the challenge maps even if they weren't tied to achievements), at which point I set a precedent requiring me to do the same for Arkham City; however, having already achieved that goal, I don't feel all that driven to repeat it on the PC version, though I still know I'm going to inevitably collect all of the 213(?) challenge medals for each of the four characters. God help me.
At any rate, my OCD is matched only by my frugality, so I always seek to finish my games (a mindset facilitated by the fact that I will virtually never lay down money for a game that I'm not already extremely confident I will like). Still, this doesn't stop me from having a by-my-standards-large Steam/GOG sale-fueled backlog, though of late I've formed the worrisome habit of starting a game and then postponing it while I switch to another (games currently occupying this state of limbo are Thief Gold, The Legend of Zelda: Ocarina of Time, Mass Effect 2, Star Wars: Republic Commando, Super Meat Boy, and Painkiller: Battle Out of Hell). I take it this "I swear I'll get to it later" mentality is what eventually leads to libraries full of unfinished games, but for whatever it's worth I do honestly intend to complete them all at some point. Time will tell.