I've been gaming since I was six years old, so about thirty years. I can remember the release of Donkey Kong in the arcades. I used to play that, as well as Galaga, Gyruss, Defender, and a bunch of others lost to the mists of time.
The first game I can remember finishing was... something I can't remember the name of. It was on the TRS-80 and written in BASIC. It was like a stripped-down version of Nethack. This would have been when I was around eight in 1983. The first game I can remember finishing that I can also remember the name of, was a vertically scrolling shooter called Exerion, in the arcades, sometime in the early eighties. Of courser, finishing a game in those days just meant it looped back to the start to play through again.
The most recent game I've finished was The Witcher 2. I'm doing another playthrough of it now, to see how a different path play out.