I think the very first time was when I finally got past the first snow stage in Turbo Outrun without running out of time
First time I played Morrowind. When you get off the boat and the world opens up to you. When you are used to only Daggerfall (which already had comparably bad graphics when it was released), the sight can take your breath away.
Finishing Dungeon Master for the first time made me feel proud of myself.
Finishing Painkiller 100% on trauma difficulty (I think it was called like that, the highest one in any case where you get the "good" ending).
Finishing Final Fantasy 7 100% for the first time - took me 80+ hours!
Finishing Super Metroid in something like 3 hours, I don't remember how fast it was precisely, but it was pretty fast. I cheated with a speed run movie though. Too bad Samus doesn't remove anything more than her outer suit
So many games gave me great feelings of accomplishment really. Super Mario 64, Banjo Kazooie, Banjo Tooie, Donkey Kong Country, etc.