  We calculate the cross sections of associated $J/\psi + \psi^\prime$ and
$J/\psi + J/\psi$ production in $pp$ collisions at $\sqrt s = 13$ TeV in the
forward kinematic region. The High Energy Factorization ($k_T$-factorization)
framework supplemented with the Catani-Ciafaloni-Fiorani-Marchesini evolution
of gluon densities in a proton is applied. We demonstrate that latest data on
$J/\psi + J/\psi$ production and first experimental data on $J/\psi +
\psi^\prime$ events taken very recently by the LHCb Collaboration can be
described well by the color singlet terms and contributions from the double
parton scattering (DPS) with the standard choice for $\sigma_{\rm eff}$
parameter. The relative production rate $\sigma(J/\psi +
\psi^\prime)/\sigma(J/\psi + J/\psi)$ is found to be sensitive to the DPS terms
as well as to feeddown contributions.
