I've been thinking a fun game would be a very simple 1v1 first person shooter where you play locally against a neural net AI whose only source of learning is playing you, and you can have it fight online against other player's AIs.
Whether that would actually be fun or not, I think NN AI in games is underutilized at the moment. Probably because running one is so much more CPU/GPU intensive than just some scripted actions for basically the same results.