While all of the planets in our solar system other than Mercury and Venus have moons, none of these moons are known to have submoons orbiting them. This is due to the fact that such objects would be affected by the gravity of the planet that the moon is orbiting. This would tend to pull the object away from its orbit around the moon and into an independent orbit around the planet. However, submoons are possible if the conditions are right.
The moon would have to be relatively large and far from the planet it orbits. Also, the submoon would have to be fairly small. Our moon, Titan, Callisto and Iapetus are all large enough and distant enough to have a submoon. Though, none have been observed.
However, space junk orbiting Earth's moon appears to be in a stable orbit. This provides evidence that a submoon of our moon could theoretically exist.
Some exoplanets have even larger moons, which are as large as Neptune. These moons orbit even larger planets. One is significantly larger than Jupiter. These moons are also very distant from their planets, and this makes them ideal candidates for having submoons.
Interesting question! I love astronomy. I used to go into the backyard with a pair of binoculars every night to look at the stars, and I actually got to the point where I could point out pretty much all the major constellations. I read every single book I could find on the topic too. Haven't gone stargazing in a long while, though. :)