Does anybody know if it is even possible for a planetary moon to have a moon of its own orbiting it?
I suppose it's possible, but I am not 100% knowing of such a moon.
Ordinary moon formation processes prevent this. However, I think it is possible for a rogue asteriod to orbit the moon. However, a rogue asteriod is far more likely to orbit the Earth than to orbit its moon. The larger the area of gravitational influence, the more likely it is to have a moon. An orbit that is at odds with a moon will eventually impact it.
To expand upon what kmarinas said, the area of gravitational influence, aka the Hill Sphere is small for moons. Moons tend to migrate after their creation. For example, Earth's Moon was originally much closer. This would have virtually eliminated any chance the Moon had for having moons of its own. Billions of years later, after arriving in its current orbit, the Moon's Hill sphere has a lot more room in it for moons, but there's no mechanism remaining for delivering a moon to the Moon. The capture method wont work.
There are no moons of moons in our solar system. However, it wouldn't surprise me if some day they were discovered in exosolar systems.
Recently several KBOs (Keuper Belt Objects, of which Pluto is the best known) have been found with multiple moons. Thus Pluto, in addition to Charon, has two other companions that orbit each other and around Pluto. [You could twist definitions a little and say that each of those is a a moon of a moon.]
I would not be surprised for a future discovery in the Kuiper Belt to properly meet the moon of a moon definition, once we can agree what a planet is! Like Tony, I am sure there are some out there somewhere, but the stability of the orbits is probably not great. One imagines them forming as the result of collisions.
The problem with moons of moons relates to the Roche lobe. Technically, the earth is a moon of the sun, and our moon is a moon of a moon [earth]. This illustrates the distance and mass scales necessary to permit stable orbits of such entities.
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