Precession of the equinoxes? For Earth, this is circa 25,000 years, although I don't believe there's a reasonably accurate measurement of this value in Antiquity.

I'm not sure that meets the criteria for a "cycle of the stars" since it's a planetary change, but I guess for Hipparchus it probably did look like a celestial change. It definitely doesn't match the claim that "we do not try to ask HOW civilizations past could have been aware", though. We know exactly how Hipparchus recognized this. Nonetheless, this is a reasonable response to my question.

Noting a long-term cycle of equinox precession is impressive but also not terribly surprising. If you can note the change, you can reasonably assume it's cyclical (because the alternative is that it's going to suddenly stop moving at some point in the future).

The Mayan Long Count calendar? If I remember right, it's method of counting leap years is more accurate than the modern Gregorian calendar.

The Mayan calendrical system uses a "year" of 360 days without attempting to account for leap days. Essentially, the long count (the classic 13.0.5.7.6 style system) is a base-20 count of the number of days from the epoch, with one of the digits being base 18 instead. The other components are the 260-day year of 20 "months" of 13 days, the 365-day year of 18 months of 20 days (with 5 extracalendrical days), (and together these two things create a 52-year cycle that's the "Short Count" calendar), a week-ish system (of 9 days), a lunar cycle count, a correction to the lunar cycle because it's annoyingly 29.5 days long, and a 819-day cycle we don't understand yet.

Their calendrical system did not attempt to correct for leap years, but we do know that they knew the length of the year to roughly the same precision as the Gregorian approximation.