Pokemon Red Version: Pokemon came out in the US when I was in 6th grade, and I was the third person in my middle school to buy it and start playing it. This was well before Pokemon became huge and I remember being motivated to wake up about 45-min earlier each morning so I could watch the anime on Fox, before it switched over to Kids WB. But anyhow, I was obsessed with Pokemon! It took me roughly a year and the purchase of a Gameshark, but I caught all 151. When Pokemon did become insanely popular I totally got sept up in the hype, seeing the first two movies in the theatre, I collected the cards feverishly, watched the anime religiously every Saturday, and bought just about everything I could afford that was Pokemon related! My love for the series seemed to die down shortly after Gold and Silver came out, and wouldn't come back until years later when Gen 4 came out on the DS.
Sonic the Hedgehog 3: My first game console was a Sega Genesis so naturally I was part of team Sega during the 16-bit era. The original Sonic the Hedgehog was my motivation to get a Genesis, however I remember hearing about Sonic 3 years later and crapping my pants! My aunt ended up getting it for me and I remember obsessively playing this game. As a kid I was only ever able to get to Icecap Zone, but that didn't stop me from replaying it over and over again. Years later the soundtrack is still my favorite of all time and it is still my favorite Sonic came hands down.
Super Mario 64: On Christmas Eve 1996 my Aunt (same one who bought me Sonic 3) surprised the hell out of me and bought me an N64 along with Super Mario 64, a game I had seen in gaming magazines and wanting so badly! I remember being absolutely floored by the 3D graphics and environment the first time I played since until then I had never seen graphics like this anywhere. It was because of this game and my subsequent obsession with it that I became a Nintendo fanatic during the 5th gen, practically ignoring the PS1 and Saturn and focusing almost exclusively in the N64 during its lifespan. Super Mario 64 was one of the first games I beat and it's a still a game I love going back to and playing all the way through.
Final Fantasy VII: I was a late bloomer to RPGs, not really having an interest in them until 2000 when a neighbor sold me on how great they are. The game that he particularly turned my on to was Final Fantasy VII. I yearned so much to play this game I actually ended up buying a PS1 along with a copy of FFVII the day the PS2 launched, which was interesting since the store employee thought I was mistaken and really wanted to get a PS2 (which i did, bit they were sold out anyways on launch day). I fell instantly in love with FFVII and to this day it is one of the only games I've put more than 200-hours into. It also made me a huge JRPG fan, especially at the time. From 2000 to 2002 I played and bought a ton of other PS1 JRPGs, and it was my favorite genre at the time. However, FFVII was always my favorite and is still the best RPG I've ever played.
Perfect Dark: Prior to playing this game I was a huge fan of Goldeneye 007, but for whatever reason this game left a far greater impression on me. I remember renting this game about 5 times before I was able to get a copy of my own, and for some reason I have incredibly fond memories of doing this. the game itself is one of the best FPS games I've ever played to this day, and I love the setting and vibe of this game so much.
Honorable mentions: The Flintstones (Genesis), Dark Colony (PC), Ocarina of Time, Donkey Kong Country 2, Sonic Adventure, Shenmue, Gran Turismo 3, Crusin World.