The game that most saddens me about not finishing is Savage Empire, an obscure Worlds of Ultima RPG made in 1990. You are the Avatar, in the service of Lord British, when you and Dr. Rafkin's laboratory are sent back in time to the time of dinosaurs in the land of Eodon, you have to find your friends and your girlfriend, unite the tribes, destroy the Myrmidex scourge, and get home. But the game doesn't work too well, or at least not with sound (and that's pointless) on the higher end Windows 95 and up systems, so once I installed Win95 in tenth grade -- 1997 -- I haven't been able to play so many great old games, at least not well. I think my copy is buggy, too.
As for current games I haven't finished, I got to one of the last parts of Xen in Half-Life and just stopped, it gets dumb; I might finish it someday, but I doubt it. I never finished any of the X-Wing and TIE Fighter games, but I want to now that I got the versions that support OpenGL graphics. I never beat Jedi Outcast, but I saw how it ends, so that's good enough for now.
On the consoles, I got to Bowser's lair in Mario 3, died, and just turned it off, so I never beat that, and I did pretty much the same thing with Scar in the Genesis version of The Lion King; got to the end, died, and realized what a waste of time it'd be to win the game.