Must Play Xbox 360 Exclusives

Although I've played a number of fantastic games on the Xbox 360, I'm of the opinion that it is a piece of crap console. (I've watched two of mine die to causes other than the red ring and am currently borrowing an extra one my friend has.) Maybe the PS3 had a lot of problems too, I don't know. But before my 360, I had NEVER had a console with this ridiculously short of a lifespan out of hardware shortcomings. (My early gen PS2 is still kicking, along with my NES, SNES and Atari 2600.)

So at this point I'm saving for a PS4 (for numerous reasons, but my issues with 360 hardware are certainly among them) and trying to play my remaining 360 games to death, squeezing the last drops of enjoyment out of them so that I can eventually sell them to help with my PS4 purchase costs.

But since I can eventually replay my favorite 360 games later if i want (either through Playstation Now streaming or PC version purchase), I want to be sure I don't miss out on any great 360 exclusives before I kick this broken box to the curb. (Metaphorically, Mark! I promise I will take good care of it and return it to you in good condition!)

Does anyone have any "must play" 360 exclusives to recommend to me or others preparing to leave their 360s behind? (Disc or download, arcade/indie, whatever.)

I just went through my 360 games and the only exclusive ones I have are for the Kinect 3 of the 5 of those ones are exercise games.
I guess the PS3 has better exclusive games or at least more exclusive games that appeal to me.

On a side note I just saw Uncharted 4 is coming out for the PS4 so might be getting one sooner then planned

I decided I am keeping my 360, but I will be getting a PS4 for some of the same reasons you mentioned.  I'm in no hurry to get the PS though, as I have plenty of games on PC and 360 to keep me entertained.  I don't generally rush through play-throughs on games, but take my time, especially for games like Skyrim and Bioshock Infinite with their amazing graphics and worlds (although I did play through all of Bioshock in under two weeks, but I've since gone back to take my time and check out more of the world).  The only Microsoft exclusive game that looked interesting to me is Titanfall, which comes out in March, but I think it is also being released on PC.  I'd also say State of Decay for 360, but I think that was also just released on Steam and seems like it'd be good to play on that platform.  With your 360 woes, it could be the same problem we discovered AFTER we picked up a new one.  Apparently it needs really good ventilation, and does get a lot of dust build-up inside which makes it totally stop working.  Check out some YT videos on cleaning out the console, just in case that is what the issue is.  It may run just fine after a little TLC.  But I do agree, it does have some short-comings compared to current and previous consoles.

The ones I'd suggest are mostly FPS like Halo and the like. However, you don't like FPS games, so that's out.

Oh oh oh oh! Geometry Wars!