Outside of the Xbox Trifecta, which is the best game exclusive to the 360?

For the sake of argument, games that were also available on PC at the time, like Witcher 2 and Alan Wake, are not included, but games that are only now multiplat via backwards compatibility or current gen remasters are still eligible.

I would love to say Dead Rising, but after 10 years, it recently got re-released on PC, X1 and PS4. Not sure it counts anymore, it did back in its day (it's not a remaster per se, it's just a re-release of the same game for these platforms, which I'm not sure it can be included on the reasoning OP gave).

There's Fable 2, which is an amazing game, full of personality (though very easy and a tad streamlined for RPG standards, even more if you compare it to its predecessor). Definitively a must if you owned a 360.

Lost Odyssey is magnificent, despite some issues it had (like being the slowest-paced JRPG ever conceived). 360 is the only option if you want to play Tales of Vesperia and you don't understand japanese or don't own a not-updated pirated PS3 and are willing to import (and even then, the extra ps3 content is still on japanese, so there's that), so I would like to include that one, but on a less "exclusive" degree than the rest given its circumstances.

Beautiful Katamari was charming, despite Bamco doing the worst microtransaction bussiness out of it I've ever seen.

All in all, my pick is the original Dead Rising, followed by Lost Odyssey if the former doesn't count, and Shadow Complex in the third place.

I'm gonna list games I personally enjoyed and sticking to the rules of 360/X1 BC exclusive...

Charlie Murder
Crackdown 1 and 2
Banjo Nuts and Bolts
Dead or Alive 4
Far Cry Instincts (ports of OG Xbox games, but maybe worth a mention.)
Fable 2
Fable The Journey
Hydro Thunder Hurricane
Lost Odyssey
PGR 4
The Gunstringer

Lost Odyssey is the king.

Otherwise I think that the first Mass Effect was a good one.

My Xbox 360 picks:
Fable series.
Kameo: Elements of Power.
Banjo Kazooie: Nuts and Bolts
Perfect Dark Zero.
Viva Pinata.
Dead Rising
Infinite Undiscovery.
Blue Dragon.
Crackdown.

Original Xbox:
Conker Live and Reloaded.
MechAssault.

Lost Odyssey.

PGR 3 and 4.
And Braid and Fez yet those are not exclusive anymore.

I enjoyed Fable 2 as well, although not as much as the first one.
The original Dead rising was great, figuring out the best order/routes to get to all the objectives in time was a fun puzzle. It's one of the few games that are actually fun to start over with prior knowledge.

Was Condemned on PS? I know it was on PC

Fable 2. I've never played Lost Odyssey but I watched some gameplay clips since everyone is praising it. It doesn't look like something I'd play.

Blue dragon.
Beautiful katamari
Condemned
Project Gotham racing
Ace combat 6

Man Xbox used to have lots of amazing exclusives.