I think some games that I really loved from my time with the Xbox 360 would be these.
Fusion Frenzy 2 is a very cool and aggressive sci-fi party game. I remember having so much fun with it, the personality and attitude games like this have sprinkled all over it is positively delightful.
Viva Pinata is oft an ignored franchise when it comes to Xbox to my knowledge but it really shouldn't be. Rare made such a fun farming sim where the quality is trying so many different things to attract different Pinatas into your garden, the first game is still pretty solid, but TIP is the perfect one to start from. It even has a 'free garden' mode which allows you to not have to deal with all the hazards in the main game since it does get very, very hectic to manage sometimes.
I fully loved this game and spent hours upon hours.
Speaking of Rare.
I love Kameo, it's a fun 3D platformer that I feel has so much beautiful parts of the early days of the Xbox 360 in one game. It has so much charm and I love its simple but charming fantasy world.
I dunno if the Arcade (the online marketplace) is still up for 360, but I love Raskulls.
It's a really charming puzzle vs. game where you try to move as fast as you can through an obstacle course and destroying blocks to get there with your staff. It's so much fun with friends and to this day, remains exclusive to the console like the other games I've mentioned here.
If you love RPG's, please try out Blue Dragon!
It was my very first 360 game and it was beautiful in so many ways back then, and I still think it has so many charming layers of fun and interesting ideas! I made a gigantic post about this game in the Xbox section of this site but yes suffice to say I adore this game like I would a mother with a child.
