2014: South Park: The Stick of Truth
The Xbox 360 is in decline now, but it still has life left in it. Many attempts were made, back in the 00's, to make a great game using the South Park license. This time, they nailed it, creating the first funny South Park game and managing to capture a lot of what makes South Park so special, without making a game that was complete nonsense.
The trappings of the story: all of the kids in South Park have taken up larping: fits perfectly with the usual setup of quests, loot drops and fantasy weirdness just slots in excellently. It's one of the few games that's genuinely made me laugh out loud (the only other I can think of is Far Cry Blood Dragon) and honestly, it's probably one of the perfect endings to the 360's life cycle.
There's a lot to love about The Stick of Truth, and the forthcoming sequel will be one to watch, too.
2015: Metal Gear Solid V
This was a tough recommendation because the Xbox 360 version of Metal Gear Solid V is definitely the worst version of Metal Gear Solid V, but it's still the best action-adventure game I've ever played.
If you've just got the 360, don't miss out on this. Dust the old beast off and experience the gift that is Metal Gear Solid V. Which, for the record, is the second highest I've ever scored a game. MGS V works equally well as either a stealth or an action game, and flying in low over hostile airspace while Ultravox's Dancing With Tears In My Ears plays from the in-helicopter speakers is fantastic.
The perfect ending to your Xbox 360's career, and a good send off in a year when basically nothing came out for the console.