I don't know what you would like, but if you only count the exclusive games, there aren't that many to choose from; most of the best 360 games are better and cheaper on Steam, or most people already have them on PS2/3 or Wii...or they've been remade for newer consoles.
There's Crackdown, if you like GTA. It's directed by the guy who did the original GTA, but you play as a cop with superpowers, so you can jump over buildings and throw cars and stuff. Like GTA, there's no real meat to the game, but it's fun in small doses.
Lollipop Chainsaw is cool if you like Suda 51.
There are a few Rare games, but I haven't played them, so I cannot give a full recommendation.
IIRC, Castlevania HD on 360 allows you to play more than one person per console, and the PS3 version does not.
There was a four-player Bubble Bobble for 360 too, I seem to remember we had a good time with it.
I guess there were a lot of games for PC that had split-screen support on the 360, if you're into couch multiplayer. Though a lot of later releases dropped it down to only two-players per console (and offline mode only).
Hmm, I'm trying to find a list of 360-exclusive games, but so far they are all filled with games that are not exclusive at all, or games that were exclusive at one point...
There were a few Sega arcade ports that were exclusive for a while, but I think you can't purchase them anymore (they were digital-only), and they're on the Switch now.
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I'll brave the storm to come, for it surely looks like rain...