Since I started gaming in the early 1980's, my first games aren't available, but I found a select few that were... and by playing with and exploring them made the gamer in me blossom at an early age.
Adventure, Atari 2600: Dragons, castles, and a sword. Oh my! Even with its simplicity, this game had alot of replayability - trying to do it faster, without getting captured, etc. This was one of my first games on the 2600, followed closely by Asteroids, but this one is special because it brought out the explorer in me... trying every niche and possibility. Oddly enough, I didn't find out about the easter egg until I was in my early 20's, but the venerable Atari still worked (and does still, last I checked), so I was able to hook it up and stroll down memory lane.
Robot Odyssey, Apple ][e: While it's listed for the PC, I actually played it on the Apple, well before the first PC was released. The challenge was to build circuits using logic gates (and, or, nand, xor), flip-flops, and sensors to make a robot explore a maze where your character cannot. It sounds simplistic (and, really, it was) but it awoke the coder in me at a very young age - about 4th grade.
Super Mario Bros 3, NES: One of my big addictions and frustrations. Due to it not having the ability to save a game, I left it on for days at a time because I could only game for a few hours. It was so much fun, though, and a fair reason while I still enjoy side-scrollers as much as I do.
Doom, PC: Aahhhh, Doom. My first foray into the FPS world and it has tainted me every since. Granted, I've never been _great_ in FPS games, but it's the speed and accuracy element that keeps me coming back. That, and the story. Granted Doom itself didn't have much story, but its successors (Doom II, Quake, Wolfenstein) did. Currently power-hearting Halo 3 ODST, the great-great-great-great-great-grandchild of Doom, imho.
Power Stone 2, Dreamcast: The Dreamcast was such a great console (VMU, hello!) but I think it was before its time. This game was one of the more fun examples from that overlooked console and could be consider the predecessor of games like Marvel: Ultimate Alliance. Frantic four-player action in dynamic stages and using props/power-ups to smack each other around with... so much fun. Then there's always the replayibility and collection aspect that wasn't in its predecessor.
This is just a taste of what my gamerDna would look like... I'd also have to include games like nethack, xpliots, and some of the newer games like Half-life (both 1 and 2) and KoToR. But I digress. Anyway, hope you enjoyed my jaunt down memory-gaming-lane.