One game I enjoy playing from time to time, especially with young students, is progressive chess. It's one of the more "normal" variants, and while there are some different versions the way I've learned it is this:
1. White starts by making one move, Black responds with two, White with three, etc.
2. A player doesn't have to give check, but once he does his turn is over.
3. When a player is in check, the first move he must make is one that gets him out of check. If he can't, he's mated.
4. All moves must be legal in the same way they would be in standard chess. (One can't, for instance, walk through a check on the way to a safe square.)
There are various benefits to this game, especially for younger/newer players. One is that it helps familiarize them with various standard checkmate patterns (especially after they've been mated with them several times) - including of course the most useful one of all for the beginner, the Scholar's Mate. Second, it forces them to pretty quickly get used to thinking not just about their own threats, but about the opponent's as well. Third, it's an excellent way to practice calculation skills (albeit a non-traditional sort of calculation). Fourth, it forces one to exercise their imagination and look for "impossible" moves - moves one couldn't normally play in standard chess. (The payoff is that one gets used to looking for "impossible" moves, which turn out to be possible and strong in standard chess more often than most players suspect.)
So I think the game is worth trying out, and it's especially worth trying out on one's own for a while. If you've never played it before, please bookmark the location of this post and come back to it after you've played progressive chess a few dozen times or more. Don't read on until then, or unless you're already a very experienced progressive chess player. Okay? Don't worry, I'll wait.
[Several weeks or even months elapse as pleasant intermission music plays in the background...]
Great, welcome back! Now that you've acquired some experience at the game and have figured out quite a bit about what works and what doesn't work, you might want to have a look at an interesting and thorough series of lectures on the game by Doug Hyatt. One caveat: by the end you might know too much about the game - it seems very nearly solved, as he discusses it - so you might want to watch one lecture at a time, only moving on to the next one after you've made whatever progress you could on your own based on the previous video.