Yeah some of the best games of each genre or series released on the Xbox. Going with OG Xbox.

Also classic games like Jet Set Radio Future, Chronicles of Riddick, Prey and Dead or Alive 3.

As a 360 fan...

Builder/Simulator: Sims 3, Viva Pinata, Minecraft... probably other stuff.
Strategy: Halo Wars, C&C, Civilization, Xcom
Open world games: Oblivion, Skyrim, Fallout, Borderlands, GTA, RDR, AC, Batman, Sleeping Dogs, Red Faction, Saints Row, Crackdown, Just Cause
Other RPGs: Diablo 3, Fable 2/3, Mass Effect, South Park: Stick of Truth, Dragon Age, Witcher 2
JRPGs: Final Fantasy XIII, Lost Odyssey, Tales of Vesperia, Dark Souls
Hack n slash: Diablo 3 (again), Bayonetta, Devil May Cry, Dead Rising,
First Person: Halo, Doom, Half Life... and Bioshock, CoD, Battlefield, Far Cry, Crysis, Left 4 Dead, Portal, Deus Ex, Dishonored, Mirror's Edge
Third person shooter/action: Gears of War, Dead Space, Resident Evil, Tomb Raider 2013 and sequel, Alan Wake
Racing: Better Forza games, better PGR games, numerous EA racers, Sonic racing games
Platformers: Motherfucking Banjo N&B, lots of Sonic, lots of indie stuff, classics like Castlevania SOTN
Fighting: Street Fighter, Marvel vs Capcom, Mortal Kombat X, King of Fighters, Soul Calibur, Tekken, better UFC games, better WWE games, Virtua Fighter 5, Fight Night, Dead or Alive

Brah, I jizzed a little making this list. I enjoyed the OG Xbox, but there was so much great stuff on 360.

I'll just end on this. If you looked at the entire library of both platforms, including the XBLA stuff, the 360 simply has more notable games.

The Xbox 360. It was really something special and innovative. Dual CPUs was not that common, and there they were coming out with a tri core processor? On a Game Machine? Crazy.

Plus, I thought Kinect was innovated (we now have Alexia and OK Google, and all those filters (ah la KinectTV Party) etc.)

I think they should have held back the Xbox One for a year and had it ready when it came out.