  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 which effectively includes some
next-to-leading order corrections. The discussed mechanisms unavoidably lead 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 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. We also present first results for the $2 \to 4$ single-parton
scattering $g g \to c \bar c c \bar c$ subprocess for the first time fully
within the $k_t$-factorization approach. In this calculation we have used an
off-shell matrix element squared calculated using recently developed
techniques. The results are compared with our earlier result obtained within
the collinear approach. Only slightly larger cross sections are obtained than
in the case of the collinear approach but still the SPS mechanism contribution
is much smaller than the DPS one. Inclusion of transverse momenta of gluons
entering the hard process leads to a much stronger azimuthal decorrelation
between $c c$ and $\bar c \bar c$ than in the collinear-factorization approach.
A comparison to predictions of double parton scattering (DPS) results and the
LHCb data strongly suggests that the assumption of two fully independent DPS
($g g \to c \bar c \otimes g g \to c \bar c$) may be too approximate.
