  We investigate whether the triple-parton scattering effects can be observed
in open charm production in proton-proton collisions at the LHC. We use
so-called factorized Ansatz for calculations of hard multiple-parton
interactions. The numerical results for each parton interaction are obtained
within the $k_{T}$-factorization approach. Predictions for one, two and three
$c\bar c$ pairs production are given for $\sqrt{s}= 7$ TeV and $\sqrt{s}= 13$
TeV. Quite large cross sections, of the order of milibarns, for the
triple-parton scattering mechanism are obtained. We suggest a measurement of
three $D^{0}$ mesons or three $\bar{D^{0}}$ antimesons by the LHCb
collaboration. Confronting our results with recent LHCb experimental data for
single and double $D^{0}$ (or $\bar{D^{0}}$) meson production we present our
predictions for triple meson final state: $D^{0}D^{0}D^{0}$ or
$\bar{D^{0}}\bar{D^{0}}\bar{D^{0}}$. We present cross sections for the LHCb
fiducial volume as well as distributions for $D^{0}$ meson transverse momentum
and three-$D^{0}$ meson invariant mass. The predicted visible cross sections,
including the detector acceptance, hadronization effects and $c \to D^{0}$
branching fraction, is of the order of a few nanobarns. The counting rates
including $D^{0} \to K^{-}\pi^{+}$ branching fractions are given for known or
expected integrated luminosities.
