No. A star is not a planet. The bodies orbiting the stars are planets, or dwarf planets, asteroids or comets. Bodies orbiting them are moons. Bodies orbiting the moons don't have a name.
At present, purely theoretical: we don’t know of any. They are probably quite rare, but we don’t really know - maybe, in centuries to come, we’ll know of dozens of examples; maybe, there are none to find in this entire galaxy
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No. A star is not a planet. The bodies orbiting the stars are planets, or dwarf planets, asteroids or comets. Bodies orbiting them are moons. Bodies orbiting the moons don't have a name.
Satellites? Natural or manmade, small or big, doesn't matter.
A natural moon of a moon is called a subsatellite: https://en.m.wikipedia.org/wiki/Subsatellite
At present, purely theoretical: we don’t know of any. They are probably quite rare, but we don’t really know - maybe, in centuries to come, we’ll know of dozens of examples; maybe, there are none to find in this entire galaxy