You need at least 1 Mario game. Be it the original, 3, Super Mario World or Super Mario 64.
Zelda also would need a spot on the list.
FPS is a little up in the air, you can go for Doom or Goldeneye or Half Life or Halo.
Blizzard needs some sort of representation on there. Any of their 3 big ones would work.
Final Fantasy as well, most would say VII.
Pokemon needs a spot, either Red/Blue or Silver/Gold.
Tetris obviously.
Grand Theft Auto needs a spot, I guess III, since it started the whole craze.
Sim something. The Sims, Sim City, etc.
That's 9 already, and I haven't even touched stuff like Civ (although I've never played) Gran Turismo, any sport game, Metroid, Metal Gear Solid, Resident Evil, and so on, and that was going really basically.
Half the reasoning behind a list like this is so you play the AAA, cream of the crop, 95%+ rated games. The other half is picking games that have had a big cultural impact, the games that define what being a gamer is.
Basically this is an impossible list.