What would the time dilation factor be if a massive (as in rest mass>0) point particle orbiting a Schwarzschild black hole in the photon sphere? If I understand correctly, this is the only possible orbit for photons, but it is also the closest possible orbit for a massive particle. So this is the same thing as asking what the maximum physically possible time dilation for a circular orbit is.
That point is 3/2 times the Schwarzschild radius. Any closer than that, and no free-fall path goes to infinity or completes a full orbit. It's also unstable. However, an exactly timed maneuver could put a particle into orbit there for a large number of orbits, so I don't think the instability affects the meaningfulness of the question. One could even realistically transition from r=infinity to the outer edge of this orbit, complete several orbits, and then escape back to r=infinity.
Reason for asking: a simplistic application of general relativity circular orbit time dilation tells me that the factor is infinity. In other words, time doesn't pass for a particle in this orbit.
I can't even begin to rationalize that. How could the universe be frozen still for such an observer? I don't think that makes any sense.