There was Phantasy Star Online released, which can't be played online anymore but still features a singleplayer portion with cutscenes and the likes. You could check that out.

Fable 2

That's about all I could think of that I played on my 360. At least, all that I played that is both exclusive and that I would recommend.

If you have a friend around or, better yet, a few and one has another xbox 360 of his own, we had sooo much fun with coop Left4Dead and Left4Dead 2.

Single player though... I doubt it's even half as fun.

Even though I haven't beat it yet myself if you like action JRPG's I would suggest Magna Carta 2.

You ever love a game even though you knew it was bad?

https://youtu.be/3YOVipx8puM

Halo, Gears,Forza, and Fable.

Lost Odyssey was better but if you like old school RPGs, try Blue Dragon.

I finished Quantum Theory, what do you think?

This looks like just the right amount of "serious" and the right amount of nonsense to me.

Exclusives I enjoyed not already mentioned already would be Too Human, Ninja Blade, and Ninja Gaiden 2. None of them are splitscreen. Too human has online co-op.

Japanese: Tales Of Vesperia, Lost Odyssey, Blue Dragon, Dark Souls?
Exclusives: Alan Wake, Crackdown, any Halo game, Forza Horizon, DOA 4, Ninja Gaiden 2
Halo: Halo Reach... I guess. I haven't played Halo 3 and I didn't like 4.