How many times can light revolve around a black hole? Take a light ray approaching a black hole from infinity which goes out again to infinity. What is the maximum finite rotation it can describe? (I know it can loop around indefinitely in the photosphere)
 A: The photon sphere lies outside the event horizon ($r_p = 3GM/c^2 > r_s = 2GM/c^2$), so a noncaptured photon can orbit a black hole as many times as it wants. Since at the photon sphere the gravitational potential of the black hole is at a peak, it is unstable, so a photon coming in from infinity and having just the right trajectory can spiral inward towards the photon sphere, but never reach it, indefinitely.
A: There are three kinds of orbits that encompass all cases: bounded $E < E_{crit}$, extreme $E = E_{crit}$ and unbounded orbits $E > E_{crit}$. As Akano has said, the extreme case can circulate around the `photon sphere' for ever, because the energy is tuned to exactly repel the inward pull. 
You can still have an unbounded orbit circulate around as many times as you like because you can take any $\epsilon >0$ and consider a photon with energy $E = E_{crit} + \epsilon$. The orbit will verrry nearly circulate the critical sphere exactly but eventually will escape. So, in theory, with infinite precision to tune the energy you could have any number of orbits ($n \in \mathbb{N}$). Of course, in reality, $\epsilon$ will have some maximum resolution (number of decimal points) it can contain, so the number of orbits will be strictly bounded.
See here (http://jetp.ac.ru/cgi-bin/dn/e_064_01_0001.pdf) for a more mathematical exposition.
A: Probably only once or twice, as a very small gravitational disturbance (a nearby electron) would send the photon into an orbit with a necessary velocity not = to C.
