Can a moon have a moon? Is it possible for a natural moon to have a natural moon? What is the limit to when satellites on satellites becomes to unnatural for it to be possible?
Of course it is possible.

If the bigger moon (A) is far enough from a parent body so that the latter one cannot pull the smaller moon (a) away from it.
So the factors at play here are the mass of the planet, the mass of the moon (A) and the distance between them. These determine what is called sphere of influence (SOI). 

So if the moon (a) orbits the moon (A) inside its SOI, it would be possible for (a) to exist there for a long time.
How long that time would be - it's a slightly more difficult question. As the tidal forces and various disturbances might eventually make the system unstable, and eventually (a) might be pulled away from (A) or even fall onto it etc.

I assume that by "unnatural" you mean "impossible"? Because usually when people talk about "non-natural" satellites, they mean "artificial", that is, "made by humans". So it has nothing to do with the type of the orbit. Natural satellites are formed, well, naturally, and artificial satellites are launched by us humans.