It really depends on the game. Usually I try to finish them. One pesky one is Twilight Princess which I got right after launch. In between that time frame I have beat countless other games but not that one. It depends on if the game either pisses me off or gets really hard and I don't want to waste my time.

In general I do. Books on the other hand are a different story. I can be in the middle of anywhere from 3 to 5 at a given time.

No, I actually have yet to finish Lost Odyssey (4th disc). It just got so boring for me that i forgot about it the moment I received Bully:SE from the mail and never looked back.

Oh yeah, and Jeane D'arc is also on my unfinished list...I played it fight after FFT:war of the lions so i was just completely burned out on SRPGs at that point.

And i also stoped playing Burnout before getting an A license. 

I used to finish every game last gen before I play a new one. I only had a GC and GBA then so it was no problem. Now I have a 360, DS, and Wii I have a stack of games unfinished or unopened.

I think I'm about to hang it up on Fire Emblem. I'm on the last battle with only one healer and I'm beginning to realize I can't win.

I try, and I will succeed.
I actually have games that are approaching 10 years old.
I still haven't finished BG II, or Neverwinter Nights (I've finished the main campaign, now I must finish the two expansions). These PC games just bore me to death, and I am an ex tabletop RPG gamer.
I'm trying to finish all my consoles RPG though. I also have other games in other genres, but I wonder if I will ever finish them, with some I just barely started :
Metroid Prime (I'll try to play this one), MGS Twin Snake (this one bore me to death), FFCC (this is a shitty ARPG, I'm not sure I'll ever finish it), several Zelda (all except LTTP that I finished, I will finish them all though), and the PS2 RPG that I bought recently and will finish for sure. There are others like God of War that I haven't even opened (but bought recently to try) or PS3 games that came bundled, like Heavenly Sword that I will finish, and F1 GP that I won't even open.
Then there are Wii games.

Wow, I could actually hold for years with the games I have already.

if its a short game then yea i usually finish them.. but longer games i usually take my time on.. unless i lose interest in a game like i have done on most GTA games...

 Given like half the events are repeats, many people gave up, though I am reasonably happy about the 7 hours I got from it (i will play it more soon)

I rarely, if ever, sit down to complete a game before I get another one.
Half the games I play and own, I never finish.

Nope, I'm literally playing 10 games at the moment that I still haven't finished. And it's only 6 days until GTA 4! Life is hard when you own a PS3, Xbox 360, PSP and DS......

I never buy a new game until I finish the previous one. I think that the truly experience a game brings is as a whole, from start to finish.

Try watching movies moving on to the next one before the one you are currently watching is over, doesn't make sense does it? sure, you are being entertained while watching but it feels like a somewhat empty experience if you don't watch a movie entirely.

I feel the same way about games. I believe they are meant to be played entirely