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I think that there is no particular reason that a moon shouldn't have a moon, however it probably wouldn't be a very stable situation, as the tidal forces from the nearby planet would tend to destabilise the orbit of the moon's moon. Essentially becasue you have 3 bodies, energy can be transferred between the two moons so their orbits can change with time.Planets are essentially moons of the sun and they have moons, so it is possible but they are a long long way from the sun so their orbits are fairly stable.
Luna is, of course, unfeasibly large, relative to its planet.
I'm not sure that Luna stabilises the Earth in any particular way
Over the last 5 million years, the obliquity of the ecliptic (or more accurately, the obliquity of the Equator on the moving ecliptic of date) has varied from 22.0425° to 24.5044°, but for the next one million years, the range will be only from 22.2289° to 24.3472°.Other planets may have a variable obliquity, too; for example, on Mars, the range is believed to be between 11° and 49° as a result of gravitational perturbations from other planets. The relatively small range for the Earth is due to the stabilizing influence of the Moon
...for example, on Mars, the range is believed to be between 11° and 49° as a result of gravitational perturbations from other planets.