  We consider the prompt double $J/\psi$ production in $pp$ collisions at the
LHC in the framework of $k_T$-factorization QCD approach. Using the
fragmentation mechanism, we evaluate the color octet contributions to the
production cross sections taking into account the combinatorial effects of
multiple gluon radiation in the initial state driven by the
Ciafaloni-Catani-Fiorani-Marchesini evolution equation. We demonstrate the
importance of these contributions in a certain kinematical region covered by
the CMS and ATLAS measurements. On the other hand, the experimental data taken
by the LHCb Collaboration at forward rapidities and moderate transverse momenta
can be described well by ${\cal O}(\alpha_s^4)$ color singlet terms and
contributions from the double parton scattering mechanism. The extracted value
of the effective cross section $\sigma_{\rm eff} = 17.5 \pm 4.1$~mb is
compatible with many other estimations based on different final states.
