It's hard to really know what counts in these kinds of discussions anymore since the question has been asked and answered so many times every response feels cliche to give. With the qualifier that both no game and almost every game end up counting by this rhetoric, I'll give the answer I always gave throughout my teens with the question came up.
Microsoft Game Studios Japan's second release, it was an unbelievable showcase for the system. Drop-dead gorgeous, gloriously weird soundtrack and mysterious narrative had me hooked when I first played it. The combat system is an action-platforming-card-fighting-game hybrid that I've still yet to see recreated or even loosely-iterated upon elsewhere in the industry. I'd like to do a full write-up on it someday because it is just that special of a game. It's team was full of a ton of new industry talent, but was helmed by veteran Yukio Futatsugi who cut his teeth working with Sega helming franchises like Panzer Dragoon.
The saddest part? Microsoft didn't even believe in the game enough to release it in the west themselves, with Majesco picking up publishing rights. Their prior game - Magatama - also deserves mention on a list like this, though it's status as a Japanese exclusive with no translation will bar many from giving it a shot. Microsoft's Japan studio would make 3 games in total, each being entirely unique from the last and dripping in inspiration and quality. Microsoft themselves didn't seem to care though and despite the 360 having a surprisingly decent launch in the region and a studio they could depend on to make great software, they shuttered them after Every Party on 360.
