1) I grew up with a Pong clone system from Goodwill, but my first "real system" was the Nintendo Entertainment System. I received it as a birthday present one year. My parents setup a scavenger hunt for me to find it. Some of my favorite games were Mario Bros. 1, 2, 3, Teenage Mutant Ninja Turtles: The Arcade Game, Mike Tyson's Punch-Out!!!, Super C, Zelda, and Tecmo Super Bowl. My favorite memory though was playing through Dragon Warrior with my mom. This was my first RPG and the only game my mom ever played. I actually beat the game with a character named "mom".
2) My grandma bought me a Nintendo 64 one year for Christmas. I played a lot of Goldeneye. I remember thinking how amazing the graphics were on that system. I was just amazed by the water and fire graphics on Turok 2. Probably my favorite game on the system was Tony Hawk Pro Skater. I remember playing level one of over and over trying to set the high score. Down the right side hill, up the quarter pipe, over the half pipe, then grind all the way around while doing grabs and flips to bump up the combo multiplier. Plus, the soundtrack was amazing, and it introduced me to Goldfinger.
3) I bought my first computer in college and played the Command & Conquer: Renegades. It was amazing! Looking back this probably wasn't the best, but my college roommate and I would start a four player game, join the same team, quickly destroy the other players base and then just mess with the other players who would join. Sniping them with "invisible snipers" was fun. Another fun prank was to buy tanks, load them up with mines, and then abandon them. The other players would sneak out all careful and "steal" the tank. Then they would get all cocky in chat, and just then we'd set off the mines, MuHaHa!
4) A rental company in my town decided to clear out their game selection. I talked with the manager and she agreed to give me first crack at the games. A few weeks later she gave me a call to come and pick out what I wanted for $2 per game, plus she gave me a 20% discount when they saw that I was buying over 100 games. I bought all of the Xbox 360, plus anything worthwhile for GameCube, PS2 or Xbox. Since then I've been buying, selling, and trading games to try to build up my collection.
5) After getting into arcade games, I finally decided to build a MAME machine. My first build was huge. It was a 4-player build with a track ball. Each subsequent version has been smaller and smaller. My current build is a 2-player game inside of a Ms. Pacman cabinet. I still enjoy playing games like NBA Jam, Street Fighter, Track & Field, and X-Men with friends or games like Ms. Pacman and Tapper when playing single player.