Anything that crosses the event horizon (including light) will swiftly proceed towards the "central" singularity (at least until it crosses the inner horizon; we will get back to that). In particular, it is impossible for an observer to hover inside the black hole. Hence any observer will be falling further down the black hole. However, it is possible for light to catch up to the falling observer. The amount of light observed in a particular time interval is however always finite, for any timelike observer.
At least this is the picture for non-spinning/non-charged black holes. Generic black hole solutions with spin or charge will feature two horizons, an inner and outer horizon. Inward motion is only necessary in the region between the two horizons. One definition of the inner horizon is that it is the boundary of the causal future of the region outside the black hole. For an observer falling into the black hole this means that as he is about to cross the inner-horizon (and he must do see in a finite amount of time), he will witness the all the light signals that entered the black hole after him for the entire future of the universe. Because the signals are compressed into a finite amount of time, both their intensity and blueshift will diverge before crossing the inner-horizon.
One consequence of this is that the naive picture of the black hole as a vacuum solution breaksdown at the inner-horizon, and we can therefore no longer trust this solution. It is expected that these divergent perturbations lead to the formation of a spacelike or lightlike singularity at the location of the inner horizon. This expectation is sometimes known as (a version of) the strong cosmic censorship conjecture. Its exact status (and the nature of the singularity at the inner horizon) is subject of ongoing scientific debate.