Probably since I was 4, so 18 years.
I remember One must fall, Blake Stone, Jill of the Jungle, Duke Nukem 2, and Galaga on a tabletop machine down by the post office.
First game I remember finishing... Probably Blake Stone or Duke Nukem but I don't remember them all that well.
Very likely Snes titles are the first I properly remember beating but I don't have a timeline of when I did so.
Last game I finished was Dragon Age 2 I guess, I haven't been shooting for completable games recently. WoW, Oblivion, and LoL have taken up most of my time, hopefully Alice Madness Returns is a proud new answer for this within a couple weeks.
Edit: RIGHT, sorry, Portal 2. (Not forgotten for lack of love of it, just misplaced my timeline)
Also, jealous of everyone older than me that clearly remembers a lot of those old ones.