  We study a large-$p_T$ three-photon production in proton-proton collisions at
the LHC. We use the leading order (LO) approximation of the parton Reggeization
approach consistently merged with the next-to-leading order corrections
originated from the emission of additional jet. For numerical calculations we
use the parton-level generator KaTie and modified KMR-type unintegrated parton
distribution functions which satisfy exact normalization conditions for
arbitrary $x$. We compare our prediction with data from ATLAS collaboration at
the center-of-mass energy $\sqrt{s}=8$ TeV. We find that the inclusion of the
real next-to-leading-order corrections leads to a good agreement between our
predictions and data with the same accuracy as for the next-to-next-to-leading
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 $\sim 15$ \% and $\sim 30$ \%, respectively.
