Probably not, at least not large moons in close by orbits. It has satellites from Earth, but they can only stay in the so called “frozen orbits” because of the Mascons - the concentrations of mass on the Moon.

​​As satellites orbit the Moon their orbits are tugged one way and then the other by the Mascons, and these keep changing the shape of the orbits. This doesn’t make them spiral down - it's not a tidal effect. Instead, it varies their ellipticity, sometimes more, and sometimes less elliptical. Eventually, in most orbits around the Moon the orbits become so elliptical they intersect the surface of the Moon and the moonlet crashes.

"FROZEN ORBITS" OF THE MOON

In an early experiment, the Apollo astronauts released a satellite PS-1 from the command module as it orbited the Moon. It lasted for over a year before it hit the Moon. But the next attempt PS-2 lasted only 35 days.

PS-2 released from the command module by Apollo 15 lasted only 35 days before it hit the Moon, although PS-1 in a similar orbit lasted over a year, Artist’s impression, see NASA - NSSDCA - Spacecraft - Details

Because of the mascoms, some orbits are more stable. And there are a few "frozen orbits" where a spacecraft can orbit the Moon indefinitely. So those are good for mission planners who want to orbit the Moon for a long time without using a lot of fuel. But would be hard for a natural satellite to get into them.

I got this from the NASA page: Bizarre Lunar Orbits where they say

"There are actually a number of 'frozen orbits' where a spacecraft can stay in a low lunar orbit indefinitely. They occur at four inclinations: 27º, 50º, 76º, and 86º"

With moonlets of moons, the tidal effect isn't that important over short timescales. And after all even the Moon's orbit is not stable indefinitely, over billions of years.

However it depends how close you are to the Moon. There’s another kind of orbit, that orbits the Moon in the opposite direction of what you expect, called the Distant Reverse Orbit or DRO. In that orbit, the moon is really in a kind of orbit around Earth, same period as the Moon but more elliptical. It could in principle even dip right down to LEO at one end of its orbit, and then go as far as that the other side of the Moon at the other end of its orbit, half a lunar month later.

By Kepler’s equal areas swept out in equal times, it would orbit Earth much faster than the Moon when it is between the Moon and Earth, and much slower when further away. So starting from a position behind the Moon at the same distance from Earth as the Moon, it would catch up with Moon on the Earth side of it, get well ahead of it, cross its orbit on the way out, slow down and then fall behind again - but as seen from Earth it is just constantly orbiting Earth in one direction.

As seen from the Moon this looks like a moon of the Moon. These orbits are far more stable than the normal orbits of the Moon and can last for at least a century, possibly longer. That’s why NASA studied them as a possible way to return an asteroid or a boulder from an asteroid to the vicinity of the Moon, safely far away from Earth.

Now from time to time Earth captures mini moons like this

Path of a simulated mini moon. It approaches Earth along the yellow path, orbits it in this complex fashion for a number of "orbits" then escapes into interplanetary space again along the red line. At any time Earth probably has at least one asteroid of diameter a meter or more in an orbit like this, along with many other smaller "mini moons". The inset shows 1999 JM8 which is much larger than any of the expected mini moons, just included because the smaller mini moons are expected to look much like their larger cousins, only smaller. From: Simulations Show Mini-Moons Orbiting Earth

Though they usually end up orbiting Earth, I don’t know if it is possible for a mini moon like that to end up orbiting the Moon, like the asteroid retrieval idea - but it seems possible - to do that it would need to get into an orbit that takes it around Earth exactly every 28 days. It would be quite hard to get into I think, because -suppose its orbit is once every 29 days for instance - it’s soon going to hit the Moon or be deflected away into some other orbit.

So I think it’s fairly unlikely, but - there are lots of small mini moons out there and we couldn’t spot them at the distance of the Moon or even if in such large DRO that they came, say, half way to Earth.

So, I’d hesitate to say it’s impossible that the Moon might have a temporary moon just a meter or so across, or stretching the definition of a moon, perhaps cms across in a DRO. Haven’t come across any paper or anything on that idea either way. But even a DRO is not long term stable so it’s unlikely to stay there for billions of years on end. At most a few centuries perhaps.

However that’s just for our Moon, because it has these huge Mascons. For a moon without mascons, it could well have a small satellite that orbits it long term. There are even ideas of places in the solar system where it may happen - well mainly its 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 (which is the mathematical limit for moons to stay long term).

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.

​Artist's impression of the rings of Rhea

This shows where it is relative to the rings and the other moons - a long way out though not as far as Titan:

Sadly, later observations to try to confirm this found no evidence of any ring system. "A very sad story": No rings for Rhea after all

But the jury is still out as to whether it has a ring system that somehow eluded discovery because those observations only gave an upper limit on the density of the ring - it could still have a thinner ring than the original observations suggested.

Also it it has these intriguing blue marks all around its equator, which may be the marks of de-orbiting ring material:

​So it might be that it forms rings from time to time. And it may or may not have a very thin ring system right now.

If Rhea does have a ring system, then just like Saturn it might have a somewehat larger (but not very big) shepherding moonlet outside of it helping to keep the material in place around the moon.

Rhea is a particularly good candidate because it is a near spherical moon (so not got the problem of irregular shape) - so if it doesn't have Mascons like the Moon, orbits could be very stable around it. And though tidally locked with Saturn with a period of 4 days, that’s not likely to have significant tidal effects on a tiny moonlet unless it has an underground ocean.

See my Robert Walker's answer to Do moons have moons?

Also my Mini moons and asteroid redirect missions

in my book Case For Moon First which is available to read free online, in its entirety, also on kindle.