I was browsing the Math Reddit today when I came across Graph War, which you can think of as an extension of the classic "Gorillas" game. In Gorillas, you were restricted to tweaking the initial conditions of a differential equation describing gravity's (and wind's) effect on a banana. Graph Wars takes it to the obvious next levels, letting you specify any function, or specify paths via first and second order differential equations. This seems like a great game for a first Calculus or Differential Equations class, especially if you let the students play with computer assistance.
Check out Graph War.