Initially, I thought this was a factor
i.e. that Saturn only travels round a thin track and repeats, but if we assume that the brighter stars are sprinkled randomly on the sky, then it doesn't matter that it repeats on more-or-less the same track.
The chance of "hitting" a star is given by the surface area of the celestial sphere (in square degrees) divided by the surface area of Saturn as seen from Earth (again, in square degrees), times the number of stars, times the distance Saturn traverses in one year.
Evidently, the answer is rather a small number...