So, I know that photons do not travel fast enough to escape a black hole once it passes the event horizon. Also, I know that the photons themselves aren't affected by the gravity, but rather their path instead. My question is, if the photons are stuck in-between the singularity and the event horizon, where do they go? Do they build up around the singularity, and they just haven't built up enough to pass the event horizon, or do they somehow escape and just don't emit light?
According to the second postulate of special relativity, light is always observed as moving at c. - However, the proper time of a photon is zero. That means, from the hypothetical point of view of the photon, no time is passing for the photon which is infalling through the event horizon towards the singularity, and afterwards emitted as Hawking radiation.
For the observers (whatever observation means in this case, because there is no direct way of measuring of photons before crossing the event horizon), in general relativity the velocity of the photon c becomes strongly decelerated in front of the event horizon, the photon is approaching (nearly) eternally the event horizon without ever reaching it.
But the photon will not approach eternally the event horizon. It will change direction when emitted as Hawking radiation. That means, from its current point near the event horizon it will be ejected outwards as radiation (from the observers' point of view).
The photons do not get "stuck" at the event horizon, from their own reference frame they are still traveling at C. The event horizon is simply the point where gravity from the singularity is strong enough that the escape velocity exceeds C. Just above the event horizon, photons can still escape the black hole, and just below it they sink into the singularity. Because the singularity is theorized to be infinitely small and of infinite density, the photons and everything else just keep falling into themselves forever.
EDIT 1: Classically the singularity at the centre of the black hole is infinitely small and the tidal forces at the singularity are infinite. This means that at some point the tidal forces will overcome the strong force and the atom will be torn apart into protons and neutrons, then quarks, then ???... Exactly what happens no one knows because we don't have a theory of quantum gravity to model it.
EDIT 2: It IS possible for a photon to get "paused" at the event horizon for a short amount of time. A photon emitted exactly at the event horizon will be in equilibrium right up until the black hole emits some Hawking radiation or swallows something, which would fluctuate the Schwarzschild radius and leave the photon either inside or outside the event horizon. If it is left outside the event horizon and escapes, it would appear as infinitely redshifted to a distant observer (it would be invisible).