I was never able to finish some games from the Hitman and Resident Evil series. Some missions seemed impossible to complete to me, so I eventually gave up. The same story with Metal Gear and Tom Clancy's Splinter Cell.
As for a game I couldn't finish it would have to be GTA San Andreas. But only being technical. You literally won't reach 100% in that game unless you manage to do everything else that is involved after you beat the missions. You need to find stuff in the water, beat all the flight school missions, and so on. Overall it even took me a month to even beat the missions. Great game though !
Well I agree with @artyarson on the Resident Evil games. I remember being so excited for the first one and then after that I could it just went away because I always got stuck midway through the game. I am not a big PC gamer and so I have not really had many games to finish lately, but I do remember a game called Myst back in the day that I never finished, and have always wondered what it would have been like.