[en] Three- and four-jet final states have been measured in photoproduction at HERA using the ZEUS detector with an integrated luminosity of 121 pb^-1. The results are presented for jets with transverse energy E^jet_T>6 GeV and pseudorapidity vertical stroke η^jet vertical stroke <2.4, in the kinematic region given by the virtuality of the photon Q²<1 GeV² and the inelasticity 0.2≤y≤0.85 and in two mass regions defined as 25≤M_nj<50 GeV and M_nj≥50 GeV, where M_nj is the invariant mass of the n-jet system. The four-jet photoproduction cross section has been measured for the first time and represents the highest-order process studied at HERA. Both the three- and four-jet cross sections have been compared with leading-logarithmic parton-shower Monte Carlo models, with and without multi-parton interactions. The three-jet cross sections have been compared to an O(αα²_s) perturbative QCD calculation. (orig.)