This month's video from Astronomic...

In theory, a moon can have a moon.  The region of space around a satellite where a sub-satellite can exist is called a 'Hill sphere'.  Outside the Hill sphere the sub-satellite would be lost around the satellite, and therefore orbiting the planet or the sun instead of the moon itself.

An easy example of this is the Sun, Earth, Moon system.  Earth is a satellite of the Sun, and the Moon is a sub-satellite of the Earth.  The Moon orbits the Earth because the earth is about 380,000km (237,000 miles) from the Earth.  This is well within the Hill sphere of the Earth, which has a radius of 1.5m km.

The Hill sphere is basically the space where Earth's gravity is more important than the Sun's gravity to another object.  If the Moon somehow ended up outside the Hill sphere of the Earth, the Moon would orbit around the Sun instead of the Earth - just like all the other planets, asteroids and comets.  

Jupiter's Hillsphere is much larger than the Earth.  That's partly because it has more mass and more gravitational gravitational pull, but especially because it is further away from the Sun, so the Sun's gravity is much weaker at Jupiter than at Earth.  This Jupiter a much bigger gravitational influence - one reason why Jupiter has a lot of moons.  Not only has it got a big Hill radius, but it's also far enough from the Sun to have its own gravitational dominance.

So if we look at the Earth's Moon, it could have a sub-satellite.  The Moon has its own Hill sphere with a radius of 67,000km (37,000 miles).  This is where the sub-satellite could exist.  If an object lies outside of the Hill sphere of the Moon it would therefore orbit the Earth instead of the Moon.  

The only problem is that the sub-satellite cannot stay in orbit around the Moon because of tides.   Like almost all the moons in the solar system, our Moon has a synchronous rotation (ie. they have the same face to their planet at all times).  This results on tidal forces between the Earth and Moon... the same tidal forces that cause high/low tides on Earth.  So any sub-satellite orbiting the Earth would have its orbit decay until eventually the sub-satellite would crash into the Moon or until be pulled apart by the Moon's tidal forces.

But in principle, the maths show that a Moon can have a Moon as long as certain
conditions are met.