Well, you have to be careful with some games ... some MtG players may end up as one of those programmers who can only write code in X language for Y purpose. :) ("No thanks, I'll stick with my red-and-black deck ...")
Certainly any game that requires some sort of programmable moves would qualify. Heck, you could throw in turn-based strategy games as well (Heroes of Might and Magic, old-school Warcraft).
But I think you could make the case that just about any game could help you with programming ... although some games might only qualify as long as you think about it from a programmer's perspective. (How would I keep the game clock running properly? How would I tell which piece the player wants to move? How do I set up an AI?) Thinking ahead, learning rules, acting only within the boundaries of the game ...
In fact, you might even try writing your own versions of games. There's nothing that shows you what it takes to write a good game quite like writing one yourself.
(Disclaimer: I'm an avid gamer. Gaming and programming have gone hand-in-hand with me ever since I could do either. When we got our Apple II+ back in the day, I was split between wanting to learn Applesoft Basic and wanting to play another Scott Adams adventure.)