  We discuss production of two pairs of $c \bar c$ in proton-proton collisions
at the LHC. Both double-parton scattering (DPS) and single-parton scattering
(SPS) contributions are included in the analysis. Each step of DPS is
calculated within $k_t$-factorization approach, i.e. effectively including
next-to-leading order corrections. The conditions how to identify the DPS
contribution are presented. The discussed mechanism unavoidably leads to the
production of pairs of mesons: $D_i D_j$ (each containing $c$ quarks) or $\bar
D_i \bar D_j$ (each containing $\bar c$ antiquarks). We calculate corresponding
production rates for different combinations of charmed mesons as well as some
differential distribution for $(D^0 D^0$ + $\bar D^0 \bar D^0)$ production.
Within large theoretical uncertainties the predicted DPS cross section is
fairly similar to the cross section measured recently by the LHCb
collaboration. The best description is obtained with the Kimber-Martin-Ryskin
(KMR) unintegrated gluon distribution, which very well simulates higher-order
corrections. The contribution of SPS, calculated in the high-energy
approximation, turned out to be rather small. Finally, we emphasize significant
contribution of DPS mechanism to inclusive charmed meson spectra measured
recently by ALICE, ATLAS and LHCb.
