  We study large-$p_T$ three-photon production at the LHC at the center-of-mass
energy $\sqrt{s}=8$ TeV. We use the LO approximation of the parton Reggeization
approach consistently merged with the real NLO corrections. For numerical
calculations use the parton-level generator KaTie and modified KMR-type
unintegrated parton distribution functions. We find good agreement between our
predictions and data with the same accuracy as in the NNLO calculations based
on the collinear parton model of QCD. At higher energies ($\sqrt{s}=13$ and 27
TeV) parton Reggeization approach predicts larger cross sections, up to $\sim
10$ \% and $\sim 20$ \%, respectively.
