Well everyone had a 360 so I don't see why not.
On the OG Ninja Gaiden and its sequel on the 360 are still up there as some of my favourite action games. I know Sigma varients came to Sony but they aren't quite the same.
There was also a pretty perfect port of Outrun 2 (does an arcade port count? it had 'only on Xbox' on the cover, anyway).
Bungie's Halo games are all fantastic.
Gears 2 is still a favourite of mine.
Lost Odyssey is the nearest thing to a more modern take on classic Final Fantasy and nails it.
There's probably loads more. I know everyone thinks of Halo/Gears when it comes to Xbox excusives but there is definitely more unusual stuff tucked away, especially on the OG.
Did anyone play Crazy Taxi 3?