From what I understand, it’s certainly possible, hypothetically, for a planet to have a moon that is both large enough and far away enough for there to be an orbit around the moon that is stable on astronomical* timescales. However, it doesn’t look like that situation exists in our solar system.
The Earth’s own moon is probably the closest candidate for this sort of thing, being fairly far away and also quite large compared to most other moons and their parent bodies. Since we haven’t seen any Moon moons, that tells us we’re not likely to in our solar system.
For more info, Google “Hill sphere.” I’m not exactly sure what the Hill radius of the moon is, but I know that for smaller satellites (eg, a weather satellites) the Hill sphere itself is much smaller than the actual object (meaning it’s impossible for anything to be in a stable orbit of the satellites).

*I almost put astrological, which would have pissed many people off no end, I’m sure But if we were talking about the Earth, I’d have said “geological” timescales. Is there such a thing as “geonomy”? Serious question

Except, umm, isn’t Dione a feminine name and doesn’t that then make it it bit rude to be talking of shaving her? What is she the bearded lady?

.. the gray lumpy potato of Epimetheus ..

Mmm .. potato! (Imagines covering Epimethus in grated cheese, maybe some carrot or pineapple and sour cream.) But I’m pretty sure that’s rock and dirty ice there instead.

@4. Trebuchet : December 13th, 2011 at 8:07 am

My first thought when I saw the first picture was “Hey, a moon with moonlets of its own!” That was before reading the post, of course. Is such a thing even possible, or does gravitation from the planet prevent it from being stable?

Well, #6 dcsohl has answered that very well already but I’ll just remind y’all that Saturnian moon Rhea possibly has not moonlets but rings of its own!

I also wonder about whether Enceladus could capture at just that right height some of the material escaping from its “tiger stripe” geysers and have that coalesce into a moonlet orbiting it or not? Mind you, if something like that has happened for Enceladus we haven’t found it yet that I’ve heard so maybe not. Too fine a source material and thus likely to disperse rather than acrete perhaps?

Then there’s the issue of co-orbital moons which aren’t quite moons of each other but are in an interesting orbital relationship with each other. It also occurs to me that we could also put an “artificial moon” – an unnatural satellite – into orbit around one of the moons.

Also asteroid 87 Sylvia has no less than three moons – but then it isn’t orbiting a planet.

Plugging in the numbers, we see that Dione’s Hill sphere has a radius of about 3250 km, which is pretty tiny especially since Dione itself has a radius of about 560 km.

Especially since the Hill sphere is only an approximation, and other orbital factors (such as other moons) can confound the dynamics. True stability can only be achieved at 1/2 or even 1/3 of the Hill sphere radius, bringing us down to the 1100-1600 km range.

So: yes, it’s possible, although more improbable things are known to have happened.

There are, for example, asteroids known to have moonlets. Asteroid (66391) 1999KW4 has a moonlet, despite the main asteroid having a Hill sphere only 22 km in radius!

So there’s hope for satellites such as Dione. But it is very very VERY improbable!

http://www.unmannedspaceflight.com/index.php?s=&showtopic=7005&view=findpost&p=181432