Yes I think so. Especially if very close to each other. If one moon is within the Hill sphere of the other moon they won’t be torn apart by tidal forces.

Our Moon is within the Hill sphere of Earth relative to the Sun. The Hill sphere is also sometimes called the “Roche sphere” but not to be confused with the “Roche limit” - if the Moon orbited close enough to Earth then tidal forces would tear it apart - that’s the Roche limit.

Similarly if two moons orbited each other close enough they would be torn apart by tidal forces, but if far enough apart they can still be within the Hill sphere but outside the Roche limit. If they are mutually tidally locked, then the Roche limit doesn’t apply, and they can be as close as one likes, even mutually touching. They then just will be deformed by each other’s gravity into a kind of a permanent pear shape as for “Rocheworld”.

So it should be possible in theory. Our Moon can’t have moonlets because of the mascoms which make most orbits very unstable and no orbits around it are stable on the very long term.

But if, for instance, the moons are reasonably uniform in composition, and spherical in shape, then they could orbit each other indefinitely.

It would be stable even in the presence of tides. The Hill sphere takes account of all of that. The gas giants have larger hill spheres. But the size of the Hill sphere depends on the mass of the satellite too. Calculation indented:

Using the formula a*(m/(3M))^(1/3) with a = 0.384 million km distance Earth to Moon, and m/M = mass of Moon in Earth masses = 0.0123 then the Hill sphere for the Moon is (0.0123/3)^(1/3)* 0.384 million km or 61,460 km. Radius of Moon is 1,737 km

This page is incorrect: Can moons have moons? (Intermediate) which comes top of the search results if you ask this question in Google. They say that the tides will remove a moon of the Moon even if within the Hill sphere. The Hill sphere already takes account or the tides, the tides are the reason that a moon of a moon would be unstable. It’s the mass cons that cause the problem for lunar satellites.

We don’t have any known examples of this in our solar system. But we do have:

CO-ORBITAL MOONS

Saturn does have co-orbital moons Epimetheus and Janus.

This is not a double moon though it may seem so from this photograph.

​Instead, they are co-orbiting and swap orbits. It is what can happen to what would otherwise be a moonlet of a moon, when the Hill sphere is too small to include both objects. So I think it is worth going into, it also helps to explain what the Hill sphere is all about.

Suppose for instance that Epimetheus is on the inside, as happens every 8 years, and Janus on the outside. Then Epimetheus is orbiting just 30 seconds per orbit faster than Janus. So it gradually gets further and further ahead of Janus until, four years later, it starts to catch up with Janus from behind.

When it does that, then Epimetheus pulls backwards at Janus - which causes it to go faster and drop into a lower orbit. Meanwhile Janus pulls forward at Epimetheus causing it to go slower and into a higher orbit. So then they swap places. Now Janus is on the inside, and Epimetheus is on the outside, and Janus will gradually speed away and so it goes on like that, swapping positions every four years. See Epimetheus (moon)

This gives a rough idea of how it works:

Only thing is they've done it with the green one much heavier so it doesn't move in its orbit.

Here is another view this time in 3D, using rotating frame:

More about it here: The Orbital Dance of Epimetheus and Janus including an actual short video of Janus and Epiphemus at the moment they change orbits (though it's taken from a perspective where it's not easy to see what is going on, a video taken by a satellite itself in orbit around Saturn and in the same plane), taken by the Cassini orbiter in 2005

Here BTW is a simulation of four moons all co-orbiting in a similar way, which could happen theoretically though no examples known, this is in rotating frame

HORSESHOE ORBITS - THE CASE OF EARTH AND CRUITHE

It's related to the idea of a "horseshoe orbit", like the much more complex orbit of 3753 Cruithne which is in a "bean shaped orbit" relative to Earth.

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From Earth's perspective:

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But something you don't see in that animation, that "bean" is also gradually drifting around relative to Earth until it catches up with Earth from behind, similarly to Janus and Epimetheus - and then when that happens, then the Earth and 3753 Cruithne do a similar swap except this time because of the difference of size, 3753 Cruithne moves over half a million kilometers while Earth moves just 1.3 centimetres.

Nevertheless that swap will move the Earth outwards a bit, enough so that your year is a little longer from then onwards, until next time the swap happens. Then it swaps back again and the process repeats. The whole process takes around 770 years.

So in this case Cruithne and Earth are orbiting the Sun. You could have a similar situation with a smaller moon and our Moon orbiting the Earth.

So anyway you can get co-orbiting moons, though they are rare.

As for moons of moons, then you get asteroids with moons and asteroids with rings. We just possibly might have moons with rings also.

One promising candidate is Saturn's moon Rhea, its second largest moon and a long way from the planet, and at one time it was thought to have a ring system, with most of it within its Hill sphere. If this was true, it would be the only moon known with a ring system, which you could think of as lots of really tiny moonlets.

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Artist's impression of the rings of Rhea

Though we don’t have any binary moons we do have

KERBEROS

Kerberos was a big surprise. they thought it was dark - to the extent, was a puzzle how it could be so dark. But turns out it is bright, and probably a "contact binary" like comet 67P.

New Horizons image of Kerberos. Its brightness is puzzling as on the face of it, it would seem to imply that it is absurdly dense, far denser than an iron meteorite, many times denser than lead. So, that's denser even than platinum. Perhaps there is something not taken account of in the gravity calculations as the author of the original paper that predicted a dark Kerberos has suggested. moonlet.

So that brings up also the idea of a contact binary moon in place of our Moon. That should work. Also a Rocheworld.

If the two planets are tidally locked to each other, there is no problem of them tearing each other apart through tidal forces so they can be so close together even that they touch.

You could replace our Earth with a Roche world like that and it could still have a Moon. Similarly you could replace the Moon with a Roche-Moon too :). You could have a “roche Moon” orbiting a “Roche world”.

Or a doublet of two moons orbiting each other like the double asteroids. There is no reason why you couldn’t also have a binary moon too. They have to orbit inside each others’ “Roche sphere” or the tidal forces of the parent body will tear them apart from each other and they will become co-orbiting moons instead - but there is no reason why that can’t happen.

Just don’t happen to have any in our solar system, at least not so far discovered. Always possible that more distant planets not yet discovered have binary moons.

See also my Robert Walker's answer to Do moons have moons - much of this is an extract from it.