Well, if you count PC games, I had an old Might and Magic game, the exact name of which escapes me since I was about 3, and Doom of course.
For consoles, I had both an NES and SNES, I don't remember which one I had first, but I do remember that I had both at the same time. Games I had for them included the Super Mario/Duck Hunt game, Super Metroid, Alien 3 (I think), Mortal Kombat 3, Killer Instinct, Donkey Kong, and quite a few others, oh yeah, and Kirby.
I moved on to the N64 later, when I saved the cash for it. I was about 9/10 when I got that and my first game for it was Kirby 64: The Crystal Shards. Of course, I got Legend of Zelda: Ocarina of Time as well.
When I traded that in, I bought a PS and had a few games for that, Judge Dredd, a few FF games that I never got into, and others.
Gave the PS to a friend for his birthday along with some games after I bought the PS2, got Primal as my first game for that I think.
Saved more money and got an Xbox, had Halo for that.
Have a 360 now, Also Halo for that, but that wasn't my first game for it, I can't recall what it was.
I have a PS3 now as well, and if I recall correctly, I had Bioshock 2 for that.
I traded in my Xbox and PS2 when I got the PS3. I just have a 360 and PS3 now.