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Protons consist of three valence quarks, two up-quarks and one down-quark, held together by gluons and a sea of quark-antiquark pairs. Collectively, quarks and gluons are referred to as partons. In a proton-proton collision, typically only one parton of each proton undergoes a hard scattering – referred to as single-parton scattering – leaving the remainder of each proton only slightly disturbed. Here, we report the study of double- and triple-parton scatterings through the simultaneous production of three J/ψmesons, which consist of a charm quark-antiquark pair, in proton-proton collisions recorded with the CMS experiment at the Large Hadron Collider. We observed this process – reconstructed through the decays of J/ψmesons into pairs of oppositely charged muons – with a statistical significance above five standard deviations. We measured the inclusive fiducial cross-section to be$$27{2}_{-104}^{+141}\,{{{\rm{(stat)}}}}\,\pm 17\,{{{\rm{(syst)}}}}\,{{{\rm{fb}}}}\,$$$272_{-104}^{+141}\mathrm{(stat)}\pm 17\mathrm{(syst)}\mathrm{fb}$, and compared it to theoretical expectations for triple-J/ψmeson production in single-, double- and triple-parton scattering scenarios. Assuming factorization of multiple hard-scattering probabilities in terms of single-parton scattering cross-sections, double- and triple-parton scattering are the dominant contributions for the measured process.