  A generic expression to compute triple parton scattering (TPS) cross sections
in high-energy proton-proton (pp) collisions is presented as a function of the
corresponding single-parton cross sections and the transverse parton
distribution in the proton encoded in an effective parameter $\sigma_{\rm
eff,TPS}$. The value of $\sigma_{\rm eff,TPS}$ is closely related to the
similar effective cross section that characterizes double-parton scatterings,
and amounts to $\sigma_{\rm eff,TPS} = 12.5 \pm 4.5$ mb. Estimates for triple
charm ($\rm c\overline{c}$) and bottom ($\rm b\overline{b}$) production in pp
collisions at LHC and FCC energies are presented based on
next-to-next-to-leading order perturbative calculations for single $\rm
c\overline{c},\rm b\overline{b}$ cross sections. At $\sqrt{s}\approx$ 100 TeV,
about 15% of the pp collisions produce three $\rm c\overline{c}$ pairs from
three different parton-parton scatterings.
