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 . The value of is closely related to the similar effective cross section that characterizes double-parton scatterings, and amounts to mb. Estimates for triple charm () and bottom () production in pp collisions at LHC and FCC energies are presented based on next-to-next-to-leading order perturbative calculations for single cross sections. At 100 TeV, about 15% of the pp collisions produce three pairs from three different parton-parton scatterings.