  We provide a detailed study of prompt double $J/\psi$ production within the
non-relativistic QCD (NRQCD) framework in proton-proton collisions at the LHC.
We confront the recent LHC data with the results obtained at leading-order (LO)
in the NRQCD framework within two approaches of the collinear factorization and
the $k_T$-factorization. We show that the LHCb data are consistent with the
$k_T$-factorized LO NRQCD results. We show that the full LO NRQCD formalism
cannot describe the recent CMS data, with about one order of magnitude
discrepancy. If the CMS data are confirmed, this indicates rather large
higher-order corrections for prompt double $J/\psi$ production. We provide
various predictions which can further test the NRQCD-based approach at the LHC
in a kinematic region that LO contributions dominate. We also investigate
long-range in rapidity double $J/\psi$ correlations. We found no evidence of a
ridge-like structure for double $J/\psi$ production in proton-proton collisions
at the LHC up to subleading $\alpha_s^6$ accuracy.
