Paradox can be simply illustrated not through a game, but by looking at a game. Take basketball, for instance. If you shoot the ball, and it goes through the net, you have scored. A paradox would be if, somehow, the ball both went through the basket and didn't. This is impossible, hence why it's a paradox. A paradox, in its simplest form, is two things that cannot at the same time be true. It's one or the other.
Literature is filled to bursting with paradoxical quotes and ideas. A game, or even an assignment, might be to assign students to find the largest number of paradoxical quotes or situations in books or stories that have been covered that semester. Alternatively you could assign them to go and find as many quotes or ideas as possible, referencing the book and page number. For instance, "Animal Farm" has the idea of all animals being equal, while some are more equal than others, and "Catch-22" takes its title from the paradoxical nature of the story's main crux.
A game that can be a real brain bender involves the paradox of time travel. You can either select an existing situation, like the H.G. Wells novel "The Time Machine" or the TV series "Doctor Who," or you can create your own with its own rules. Pick a situation where an incident of time travel happens, and then decide whether or not it's a paradox. For instance, the idea of time travel to the past might be a paradox, because the past has already happened and alterations to it could stop time travel from happening in the first place. Time travel to the future, on the other hand, requires you to muck around with potentials, and not with things that happened before your creation, so it may seem more possible. You could set this as a debate style, with one person on either side of the issue, or you could do it as a discussion in the round, moving from person to person to discuss paradox.
One game that can show paradox involves taking a cue from the medical field. Two players take on the roles of doctors, and a third player takes on the role of a patient. The patient displays some symptom, and the doctors have to figure out what the disorder is. This can be a great example of paradox because if a patient has one disorder -- such as depression -- then he won't display symptoms of another disorder, such as the manic phases of someone that's manic depressive. To keep it simple the disorders should all be chosen first, so that all players know what the options are and what they're looking at.