NES: I was in... second or third grade, maybe? Anyway, the games were way too intense for me—my mom likes to tell the story of how I would sit in a racing arcade cabinet and when my car crashed, I fell out of the machine myself. I was definitely one of those kids who physically jerked the controller around. Playing the games myself was too intimidating, so I mainly remember watching my dad play Adventures of Lolo.
Gameboy: The first few years were dominated by Tetris. After the SNES came out, Link's Awakening would take over.
Super Nintendo: Squaresoft. I wouldn't deliberately put a game in and start playing by myself on a tv-based console until my friend couldn't come over to help me finish Secret of Mana. Final Fantasy VI made me love the company, and I experienced my first game anticipation waiting for Chrono Trigger, Secret of Evermore, and Super Mario RPG. And Earthbound, even though it wasn't Square, because of Nintendo Power's ridiculous marketing campaign.
Nintendo 64: Nintendo's first party multiplayer games. Mario 64 was the first platformer I was able to finish myself, but most of my time was spent with Goldeneye and Super Mario Kart. I'm sure I finished Goldeneye, but I have no memory of doing so; I spent far more time wandering the multiplayer maps by myself, sometimes setting up stationary targets with extra controllers, than I spent in single player. It really paid off when my Boy Scout troop had a Goldeneye tournament, since I won.
PC: Final Fantasy VII, played at 1/4th the size of our ancient monitor's maximum resolution. I felt betrayed that Square had jumped ship from Nintendo, and spent the year before VII's release complaining about how stupid Cloud looked, but as soon as I played it at my friend's house, I had to have it.
Playstation: Final Fantasy VIII. My interest in Squaresoft proved stronger than my interest in Nintendo.
After Chrono Cross came out, I entered college--when I realized I could use my college loans and the money I'd earned to actually buy consoles and games for myself. My game-buying was too frequent for any single game to stand out, but I'll try to come up with one, or at least a category.
GBA: Final Fantasy Tactics Advance
Playstation 2: Silent Hills, Final Fantasies, Grand Theft Autos, Metal Gear Solids
Gamecube: Super Mario Sunshine, and things that used the GBA/GC cable (Animal Crossing, FFCC, Zelda: Four Swords)
Xbox: Psychonauts and (sigh) Fable
Nintendo DS: Wow. Uh. Phoenix Wrights, probably.
Xbox 360: Really hard to say, but Halos and Gears of War, mainly.
Wii: Super Smash Bros. Brawl is about to get overtaken by Wii Fit.
Playstation 3: Little Big Planet, thus far.
PSP: Metal Gear AC!D, maybe? Dissidia and Crisis Core are close.