When/where does gamma-gamma scattering occur and what would it look like on a large scale? I was listening to an interview with Brian Cox and he mentioned that gamma-gamma scattering is when two photons "bounce off" each other and it occurs at "sufficiently high energy."
What sort of "high energy" are we talking about? Meaning, is there any natural occurrence in nature (say something like quasars) that is "sufficiently high energy" enough to cause photons to interact with each other?
On a large enough scale—say two ultra-high-energy, cosmic-scale flashlights—what would be the effect? Would many of them interact at such high energy, or would only a small fraction of them have any noticeable effect? Would it be visible to the human eye?
 A: The scattering $\gamma \gamma \rightarrow \gamma \gamma$ is a loop induced process. Which means it does not occur at tree level pertubation order. Therefore it is not that likely to occur, but it can e.g. by processes including a box of charged leptons running around.
Some calculation of the cross section for the process is given in this paper by Böhm and Schuster (Scattering of light by light in the electroweak Standard Model), which should answer the question in which energy scales light by light scattering occurs. This paper should be interesting to you since they calculate the mean free path (due to scattering on the cosmic microwave background) for low energy photons.
More recently light by light scattering was observed at heavy ion collisions at CERN, where they collide Pb-ions with which, caused by there high charge, photon photon scattering is detectable. You can take a look at this paper.
The process is kind of interesting because it could be sensible for beyond standard model extensions including e.g. charged higgs bosons.
A: Light dominantly interacts with objects that have electric charge. Since light itself does not have electric charge, one photon cannot directly interact with another photon. Instead, they just pass right through each other without being affected. Because they are bosons and because they carry no electric charge, one photon cannot directly bounce off another photon. For example, when you point two sources of lights (say, flashlight) at each other, the light rays simply pass through. In contrast, if you point two water jets at each other it becomes a mess at the intersection point.

However, two photons heading towards each other can indeed collide indirectly. The process goes like this. A photon can spontaneously degenerate into a particle with mass and its antiparticle in a process known as pair production. In this process, the energy of the photon is completely transformed into the mass of the two particles. For example, a photon can turn into an electron and an anti-electron. If two photons head towards each other and they both turn into electron/anti-electron pairs at about the same time, then these particles can interact. The anti-electron from one photon will collide with an electron from the other photon, and turn back to light. The same thing happens to the other anti-electron and electron. The overall effect is that you get two photons going into the interaction and you get two photons coming out of the interaction, so it looks like the photons simply bounced off each other. In a sense, the one bit of light did indeed bounce off the other bit of light, but only indirectly by transforming into other particles.

Only gamma-rays can collide indirectly. And gamma-rays are just "high energy" photons. Pair telescopes work on this principle.

Photon-photon scattering is therefore possible through an indirect mechanism, but it is rare. There are two reasons that it is rare. First, light can only turn into other particles if it has enough energy to create the mass needed for the new particles, according to E = mc2. Because c is such a huge number, it takes a large amount of energy to make a little bit of mass. In order to turn into an electron and an anti-electron, the photon must have at least as much energy as the equivalent energy of their combined masses. Only gamma rays (one step higher than X-rays) have enough energy to do this. Secondly, the photons have to transform at just the right moment in order for the new particles from both photons to collide. For both of these reasons, light-light scattering is very rare. In fact, light-light scattering has never been conclusively observed. All the steps in light-light scattering have been observed (pair production and pair annihilation), so that we know that it is possible. The whole effect is just so rare that it has never been observed.

Source
As pointed out by Beliod, light-by-light scattering has been observed. However, do note that this was an experimental observation and not because of natural causes. Therefore, the last part saying "The whole effect is just so rare that it has never been observed" still is correct.
