  We study double prompt $J/\psi$ hadroproduction within the
nonrelativistic-QCD factorization formalism adopting the parton Reggeization
approach to treat initial-state radiation in a gauge invariant and
infrared-safe way. We present first predictions for the cross section
distributions in the transverse momenta of the subleading $J/\psi$ meson and
the $J/\psi$ pair. Already at leading order in $\alpha_s$, these predictions as
well as those for the total cross section and its distributions in the
invariant mass $m_{\psi\psi}$ and the rapidity separation $|Y|$ of the $J/\psi$
pair nicely agree with recent ATLAS and CMS measurements, except for the
large-$m_{\psi\psi}$ and large-$|Y|$ regions, where the predictions
substantially undershoot the data. In the latter regions, BFKL resummation is
shown to enhance the cross sections by up to a factor of two and so to improve
the description of the data.
