The Game within the Game

Chase the Queen
This is a favorite of ours. It's a great way to punish those who rush their queen out early. You may be thinking, but strong players use the Scandinavian Defence (1.e4 d5, 2.ed Qd5), and you'd be right. And one of the variations for White includes chasing the Queen around. And against anyone below titled level, I highly recommend playing "Chase the Queen!" Even if they play the Scandanavian, the concept is still valid. Develop your pieces and occupy the center (with pawns) while you chase your opponent's queen. This means you gain tempi, develop faster, and should be prepared to pounce.

A secondary advantage is that occasionally you'll actually trap the queen. Who doesn't like winning a queen?

Chase the King
Another game, a better game, is Chase the King! It's better because;

1. Unlike in Chase the Queen, your opponent has to deal with the attack (they could leave their queen en prise, especially if they can attack your king)
2. If the King is on the run you gain all the benefits of Chase the Queen (tempo grabbing development and center control)
3. Occasionally you'll end up checkmating your opponent

NOTE: in both of these cases, you should be playing the Chase Games to improve your position and expedite development, gain space and time. Don't just check or attack the Queen to do so...especially if you are moving your pieces too many times, not improving your position and losing time.

Guess the move
This game doesn't give you an advantage on the board like the Chase Games do, but it does give you an advantage in thinking. Perhaps you have done your calculations, you feel good about your position, and you're waiting for your opponent to move. Perhaps you just can't get into thinking three moves ahead at the moment. Here's a game you can play to break that block or get out of a rut.

Try to guess your opponent's move. It's great if you start by figuring out their best move...but then try and guess which move they will actually make! If you're right, you get a point. See how many points you can get. And while you're playing, you can figure out what you'll do if you were right, and save time on your next move.

Tempo, tempo, who has the tempo?
Our last game is to play with a purpose in mind: can we get/steal tempos from our opponent? See if you can chase pieces (any piece, not just Queen or King), take squares away, encourage bad moves, while making good moves in return. Like retaking with an undeveloped piece rather than on that is already in a good square.

Can you think of any "games within the game" you can play?