<p>We compute the photon self-energy to three loops in Quantum Electrodynamics. The method of differential equations for Feynman integrals and a complete <i>ϵ</i>-factorization of the former allow us to obtain fully analytical results in terms of iterated integrals involving integration kernels related to a K3 geometry. We argue that our basis has the right properties to be a natural generalization of a canonical basis beyond the polylogarithmic case and we show that many of the kernels appearing in the differential equations, cancel out in the final result to finite order in <i>ϵ</i>. We further provide generalized series expansions that cover the whole kinematic space so that our results for the self-energy may be easily evaluated numerically for all values of the momentum squared. From the local solution at <i>p</i>² = 0, we extract the photon wave function renormalization constant in the on-shell scheme to three loops and confirm its agreement with previously obtained results.</p>