The production of six energetic jets in proton-proton (pp) collisions at the LHC is studied as a means to experimentally observe for the first time the simultaneous scattering of three partons. The single-parton-scattering (SPS) cross sections for the production 2-, 3-, 4-, and 6-jets in pp collisions at center-of-mass energies of $\sqrt{s}= 14$ TeV, are calculated up to next-to-leading-order (NLO) accuracy in perturbative quantum chromodynamics with the MadGraph5_aMC@NLO
and Alpgen codes complemented with Pythia-8 for parton showering, hadronization, and decays. Jets are reconstructed using the anti-$k_T$ algorithm with distance parameter $R=0.4$. Assuming factorization of multiple hard-scattering probabilities in terms of SPS cross sections, the contributions to six-jet production from double- (DPS) and triple- (TPS) parton scatterings are derived. We find that the TPS contributions represent a ${\approx}20\%$ (${\approx}1\%$) fraction of the total 6-jets yields for minimum jet transverse momenta of $p_{T,min} = 20$ ($40$) GeV. A detailed multivariate analysis with realistic simulations of fully reconstructed jet samples for the TPS signal and DPS and SPS backgrounds indicates that TPS can be observed in events with six jets with $p_{T,min}=40$ GeV each, by collecting an integrated luminosity of $\mathcal{O}(50$ pb$^{-1}$) in a dedicated low-pileup run at the LHC.