1. Alien Soldier
Discovered this while trying to complete my Treasure collection. Treasure is my favorite developer, and this is my favorite game. In my mind, it's the closest thing to a perfect game there is: blistering action, simple concept but complex tactics, and extremely rewarding.
2. Gunstar Heroes
One of the earlier games I've played, it has sentimental value for me because I beat it one awesome afternoon with my Dad. I do the occasional play through usually once every two months.
3. Kirby's Dream land
This was the first game I've ever beaten. Not much more to it then that.
4. Ragnarok Online
My very first online game was also my first MMO. I was not prepared. I played this my freshmen summer in high school. It destroyed me. I remember one specific instance where I was playing at 4 or 5 in the morning and nodded off. When I woke up around 7, I was a level higher then my last memory. I decided to get some sleep after that.
5. Final Fantasy IV
My first RPG. I actually played the PS1 re-release version (loading times and all). I remember the only reason I bought it was to impress the girl across the street who was into RPGs (I had no experience with them previously). I think I became a true "gamer" after this RPG baptism.
6. Metal Gear Solid
The first "serious" game I enjoyed. It was also the first game that made me think (the ones before where usually mindless platformers or other action games). Because of this, I tend to overindulge the philosophical aspects of the MGS series.
7. Persona 4
The most recent game on the list, I played this during a pretty shitty time in my life. I wasn't employed, wasn't going to school, and had pretty much given up on life. Though I'd be hesitant to say Persona 4 inspired me to get out of this funk, it definitely nurtured me during a possibly dangerous part of my life. Also, it was a great friggen game.
8. Guilty Gear X2
Though not my first fighter, it was the first game that I played semi-professionally. I had daily practice sessions and went to a lot of local (and some not so local) tournaments. It was also the first game that I broke a controller on (fuck you, Faust).
9. Commander Keen
The earliest gaming memories I have are of this unassuming DOS game. I really should play through it again sometime.