I just finished Bioshock 2 for the first time. I did Bioshock twice, which in itself is a very rare thing since I seldom finish a game before my interest wanes. Some games have gotten too hard but most often I just lose interest or my computer will crash and I don't want to replay a lot of the same stuff over again.
Yeah, I'm only ever really able to afford moderately powerful PCs and so I get barely two years of gaming with contemporary stuff before I have to start raiding the budget titles again. In some ways it's a moderating influence on my play, not only is it cheaper but I get to rediscover old classics as well, and nothing sweeter than pleasure delayed. So yes, for me, computer issues is the main reason I can't finish a game. (My computer seems to start accumulating hard disc errors every time I have temerity to use the thing for data storage - the craziness!)
Then again, the problem with resorting to the bargain bin is one inevitably splurges and ends up with LOTS of games. So as much as anything else, I think I'm suffering from too much choice. I've gotten a lot more easily distracted as my collection has expanded. I'm the same with books (although I tend to reread books seldom)....
How important is it to you to finish a game? Is there some OCD drive sometimes or does it matter if you ever finish a game?
It depends what you mean by "finish". If it's a game with a considerable single-player component with a good storyline and interesting gameplay, then yes, I will typically finish the storyline.
The thing that just doesn't appeal to me is "completion". I'm replaying GTA San Andreas at the moment, and the levels of detail one has to play sections of the game to in order to get 100% COMPLETION (with no hint of unlockables) staggers me. I just don't get that level of replayability. I think for me part of it is if I'm finding a gameworld particularly immersive, the notion of "completion" seems to me to be a massive neon sign saying "Well, you hit the limits of the little pretend world you've had so much larking about in. Back to the real world you go!" If a world is immersive, I like the illusion to last.
If there's some unlockables involved, that might get my attention - but even then, it's touch and go. Best games are those I want to replay anyway and still have unlockables
Does it depend on the story? That was one of the key factors in me finishing Bioshock twice and now Bioshock 2, the story was so compelling. Granted the fatherhood element was really working me since my son is small but there were times when I was really moved emotionally by turns in the story.
I think I get hooked on story to a greater extent than yourself, I tend to replay games nine or ten times through if they really grab me. I've probably played through Deus Ex even more than that.
(Bioshock 2 definitely gripped me, especially towards the end. Bioshock 1 was a tough act to follow though, and some of the flaws in 2 have made me less keen on replaying it. The last few levels are solid gold, though.
Getting excited about BioShock Infinite yet?

)
What are some games that you want to finish but have never gotten around to?
Starcraft 2 (thank you, hard disc errors....and possibly the first game in a while that I actually really wanted to get on the multiplayer ladders for)
Ditto Batman: Arkham asylum (computer issues)
Ditto Mirror's Edge (I <3 the UT3 engine...even though it abuses my graphics card)
Operation Flashpoint (because it's as hard as nails and the save system is ridiculous. I'm amazed I got as far as I did, think I'm only a couple of missions from the end)
Rome: Total War (brilliant game, but I absolutely FAIL at it. Have yet to win a game. I'm great in the tactical battles, but rubbish strategically in the turn-based world map stuff)
Ditto Europa Universalis 3.
Ditto X-COM series.
Fallout 3 (I basically had heard that the excellent single-player content was rather short, so I stopped before the last mission, got the level 20 explorer perk and set off into the game world to see what's what. Never technically finished the single-player stuff though.)