  Multi-photon exchange contributions are important in extracting the proton
charge radius from elastic electron--proton scattering. So far, only diagrams
associated with the exchange of a second photon have been evaluated. At the
very low $Q^2$ values relevant to the radius extraction, higher order
contributions may become important. We evaluate these corrections in the
Effective Momentum Approximation, which includes the Coulomb interaction to all
orders, and find small corrections with a strong $Q^2$ dependence at low $Q^2$
and large scattering angles. This suggests that the higher order terms may be
important in the evaluation of the proton magnetic radius.
