Considering these are must haves, I'll do one per genre:
RPG-The World Ends With You
Strategy-Advance Wars: Dual Strike
Simulation-Anno 1701
Puzzle-Professor Layton and the Curious Village
Platformer-New Super Mario Bros.
SHMUP-Contra 4
2D Action Adventure-Castlevania: Dawn of Sorrow
3D Action Adventure-Grand Theft Auto: Chinatown Wars
Adventure-Hotel Dusk: Room 215
Racing-Mario Kart DS
Sports-MLB Power Pros 2008
Fighting-Bleach: Dark Souls
Music/Rhythm-Elite Beat Agents
This is actually 13, but I didn't want to cut a genre out just to concede to the 10 game rule. My personal favorites differ a bit, but I tried to use the games more generally accepted/popular and not include my niche favorites. There would be a few changes though if I put more of my personal twist into things.