A few hundred kilometers beyond geosynchronous is the satellite graveyard orbit, which distance is also a function of satellite size. One could place an asteroid into such a orbit to be a quite visible moon. One can argue that since we do not have such a moon, that it is therefore improbable. However, as below, no moon is forever, and most assuredly in the past we had lots of little moons. Earth more or less has a sort of a second moon, and will for centuries so "can't have" is doubtful.
I do not buy the "I do not like the complicated orbit small temporary natural satellite argument." Using that argument, Phobos would not be a moon of Mars as its orbit will terminally decay in 30 million years. 196
Our Luna will eventually escape due tidal braking, so, I would prefer to say that no satellite is forever. The next problem is that if 2016HO3 above is not a 'moon' or 'satellite' then Earth is not a 'planet,' as planets 'clear their own orbits' as point 3 of the IAU definition of planet.
The Hill sphere is only an approximation, and other forces (such as radiation pressure or the <PERSON> effect) can eventually perturb an object out of the sphere. This third object should also be of small enough mass that it introduces no additional complications through its own gravity. Detailed numerical calculations show that orbits at or just within the Hill sphere are not stable in the long term; it appears that stable satellite orbits exist only inside 1/2 to 1/3 of the Hill radius. The region of stability for retrograde orbits at a large distance from the primary, is larger than the region for prograde orbits at a large distance from the primary. This was thought to explain the preponderance of retrograde moons around Jupiter; however, Saturn has a more even mix of retrograde/prograde moons so the reasons are more complicated...The Hill sphere of (66391) 1999 KW$_4$, a Mercury-crosser asteroid that has a moon (S/2001 (66391) 1), measures 22 km in radius. 196