  We compute the potential-photon contributions to the classical relativistic
scattering angle of two charged non-spinning bodies in electrodynamics through
fifth order in the coupling. We use the scattering amplitudes framework,
effective field theory, and multi-loop integration techniques based on
integration by parts and differential equations. At fifth order, the result is
expressed in terms of cyclotomic polylogarithms. Our calculation demonstrates
the feasibility of the corresponding calculations in general relativity,
including the evaluation of the encountered four-loop integrals.
