I just mentioned Conway's Game of Life as an example of a Turing-complete system.
Though it's called the "Game" of Life, and some people describe it as a zero-player game, it's not really a game in the traditional sense of the word.
For an unexpected example of an unambiguous game which is also Turing-complete, check out Magic: The Gathering. It has been shown that a Universal Turing machine can be constructed inside Magic, and hence that it is Turing-complete.
I played a lot of Magic in high school, though it's too expensive of a habit to keep up for long. If, like me, you remember Magic fondly, but don't play it so much any more, might I recommend Mark Rosewater's Drive to Work podcast?