Off the top of my head and sticking with Windows based PC games and X-Box (I can't remember all of my old Commodore games
)...
Baldur's Gate 1 & 2 (probably around 10 times and still going)
Mass Effect 1 & 2 (probably about 9 times and still going)
Dragon Age (about 8 times...still playing)
Dragon Age 2 (5 times...still playing)
KOTOR 1 & 2 (probably 8 times or so...still may play 2 a few more times with mods)
Vampire Redemption (probably 10 times or so)
Vampire Bloodlines (about 6 times)
Neverwinter Nights (about 6 times)
Neverwinter Nights 2 (about 3 times)
Icewind Dale 1 & 2 (about 6 times)
Betrayal at Krondor (maybe 3 times)
Return to Krondor (maybe 3 times)
Realms of Arkania:Star Trail (5 times or so)
Diablo 1 (5 times or so)
UFC 2010 and 2011 (I consider finishing reaching Hall of Fame status with a career fighter, which I did in both games once)
Does finishing career mode in games like Rock Band and Guitar Hero count? If so, then I did it in all of those games (RB2, RB3, GH 3-6 + GH Metallica).
Brutal Legend
Star Wars: The Force Unleashed
Gabriel Knight (about 4 times)
Gabriel Knight 2
If winning the Stanley Cup counts in the EA Sports NHL series, I've accomplished that several times on each year's version.
Man, I know I'm forgetting several games. Maybe I'll check my game stash when I get home and add a few more. As you can see, I often replay games I really like...particulary Bioware games.
I normally finish games unless I'm really not enjoying myself (like Diablo 2).
(EDIT - Okay, here are a few more games that I forgot...)
Left 4 Dead (many times)
Magic the Gathering - Duels of the Planeswalker (5 times or so)
Jade Empire
Wizardry 8 (3 times)
Realms of Arkania: Shadows Over Riva (5 or 6 times)
Might and Magic VI (2 or 3 times)
Might and Magic VII (2 or 3 times)
Might and Magic VIII (2 or 3 times)
Might and Magic IX (2 or 3 times)
The Temple of Elemental Evil
Dungeon Siege
Sacred
The Elder Scrolls 2: Daggerfall
Divine Divinity