Your question is a very good one. A moon of a planet could have its own moon if the gravitational conditions were right. The problem is that the gravity of the planet that the main moon is orbiting is usually large enough that the moon's moon would be pulled away and become a moon of the larger planet. To my knowledge, there are no moons of moons in the solar system. There are several cases around the outer planets where two or three moons share the same orbit around the main planet. Do you think
these moons might have orbited one another in the past? There is at least one case of an asteroid orbiting another asteroid where the smaller asteroid Dactyl orbits a larger asteroid named Ida. In this case, there is no nearby planet to disrupt the gravitational attraction between these asteroids. You can learn more about the asteroids at NASA's website http://sse.jpl.nasa.gov/features/planets/asteroids/asteroids.html
Follow the links back from this site to learn more about planets and their moons.
I hope this helps with your question.

Absolutely. We know that several asteroids actually have tiny moons. The earth has many moons... all but one is man made!!! Basically when two bodies have a close encounter ONE OF THREE THINGS can happen:

1) They can collide
2) One body will WHIP by the other narrowly AVOIDING a collision and get sent on an orbit which will take it away from the other body forever
3) The two bodies can go into stable orbit around each other. We say that the smaller of the two bodies has been CAPTURED by the larger one.

When you actually sit down and work out the celestial mechanics of this business you find out that CAPTURE is RARE!!! Most of the time the bodies either collide or after making a close pass never see each other again.