  We present the complete next-to-next-to-leading order (NNLO) pure pointlike
QED corrections to lepton-proton scattering, including three-photon-exchange
contributions, and investigate their impact in the case of the MUSE experiment.
These corrections are computed with no approximation regarding the energy of
the emitted photons and taking into account lepton-mass effects. We contrast
the NNLO QED corrections to known next-to-leading order corrections, where we
include the elastic two-photon exchange (TPE) through a simple hadronic model
calculation with a dipole ansatz for the proton electromagnetic form factors.
We show that, in the low-momentum-transfer region accessed by the MUSE
experiment, the improvement due to more sophisticated treatments of the TPE,
including inelastic TPE, is of similar if not smaller size than some of the
NNLO QED corrections. Hence, the latter have to be included in a precision
determination of the low-energy proton structure from scattering data, in
particular for electron-proton scattering. For muon-proton scattering, the NNLO
QED corrections are considerably smaller.
