Multi-electron production is studied at high electron transverse momentum in positron- and electron-proton collisions using the H1 detector at HERA. The data correspond to an integrated luminosity of 115 pb-1. Di-electron and tri-electron event yields are measured. Cross sections are derived in a restricted phase space region dominated by photon-photon collisions. In general good agreement is found with the Standard Model predictions. However, for electron pair invariant masses above 100 GeV, three di-electron events and three tri-electron events are observed, compared to Standard Model expectations of 0.30 \pm 0.04 and 0.23 \pm 0.04, respectively.