Ok, I managed to make a TENTATIVE list. I reserve the right to change my mind later.
Super Mario Bros/Duck Hunt
First game I ever owned. I'm sure it's like that for a lot of people. It all started here, baby.
Final Fantasy
First RPG. Up until I played this all I really knew of gaming was running and jumping and shooting. Those are all fine, of course, and I still love doing all that, but this is the game that really opened things up for me and gave me a game that challenged my ability to strategize instead of my reflexes. It made me realize I love RPGS.
Chrono Trigger
I didn't actually play Chrono Trigger until after the SNES was obsolete, but it did so many things that were still very unique at the time, and this was the game that made me really care about the characters. It was a tough call between this, FFVI, and Phantasy Star IV, but the gameplay mechanics give this one a slight edge. It all blew me away.
Sonic 3
I had Sonic 1 and 2, but this is the first one I played over...and over...and over and over and over and over and over and over. Sonic 3 was practially a religion for me for a good while. Having a save file (unusual for a game like that at the time) made all the difference.
Vectorman
This is the first game I REAALLY drooled over as a kid. I wanted it more than anything. The ads made it look so amazing, Sega built up all kinds of hype, and I think I actually dreamed about it a few times. This was my personal intro to aggressive advertising by a game publisher. I never did own it, although I did get to rent it. I was probably looking through rose-tinted glasses at the time, but I just felt this sense of fulfillment in finally getting to experience this game that I obsessed over.
Final Fantasy X
Whether you liked or hated FFX really doesn't matter. This was my first PS2 game, and when I saw that opening cutscene I was in awe. This was my personal proof that the next generation of gaming had arrived.
Doom
My first FPS and I am pretty sure my first M-rated game. Primitive by today's standards, but blasting demons with a shotgun was something I could and did do for hours on end. I think my testicles may have descended when I finished the first shareware episode.
SimAnt
SimCity was too complicated for me and I had a crappy PC, so for a long time this was my primary (only?) PC game, and I was fine with that. I learned more about ants than I will ever need to know. If you know what you're doing you can beat this game in about 30 minutes or so, but that didn't stop me from playing it over and over. The only reason I stopped was because my next PC wasn't compatible with it.
Day of the Tentacle
Not only did this make me fall in love with adventure games, but it showed me just how much difference a sense of humor can have. Without the fantastic writing, Tentacle would be an okay game, but not great. Frankly, I think more games could benefit from a well-crafted joke in just the right spot. So many games these days are super srs bsns. I like games that don't take themselves so seriously.