Do photons slow down as they approach the event horizon of a black hole? So, if a black hole is so dense, that light cannot escape, what happens to the photons as the approach and go past the event horizon (or some other horizon, about which I am ignorant)?  Do photons slow down at some point, or is it somehow binary, where at one moment they are moving at the speed of light and the next they aren't moving at all?  Or, do they continue to move at the speed of light, just never moving beyond the black hole?
Photons travel on null geodesics through space time. In the curved space-time near a non-spinning black hole, a distant observer could infer that the speed of light (travelling on a radial path) is given by
$$ v_{\rm light} = c \left(1 - \frac{r_s}{r}\right),$$
where $r_s$ is the Schwarzschild radius.

As $r \rightarrow r_s$ then we see that $v_{\rm light} \rightarrow 0$. So yes, according to an observer far from a black hole, the speed of light approaching a black hole appears to slow down and asymptotically approaches zero at the event horizon.

However, the locally measured speed of light is always $c$.