We consider a model of massive scattering amplitudes in N = 4 super Yang-Mills, which is analogous to light-by-light scattering. Working at the planar level, we provide the full three-loop amplitudes. We then derive various asymptotic expansions in physically interesting regimes. We find a surprisingly simple structure in the Regge limit, and suggest a simple form of the first power-suppressed terms. We propose that the latter are controlled by an anomalous dimension, and provide perturbative evidence for this conjecture.