Multi-Leptons with High Transverse Momentum at HERA (H1+ZEUS combination) Events with at least two high transverse momentum leptons (electrons or muons) are studied using the H1 and ZEUS detectors at HERA with an integrated luminosity of $0.94$~fb$^{-1}$. The observed numbers of events are in general agreement with the Standard Model predictions. Seven di- and tri-lepton events are observed in $e^+p$ collision data with a scalar sum of the lepton transverse momenta above $100$~GeV, while \mbox{$1.94 \pm 0.17$} events are expected. Such events are not observed in $e^-p$ collisions for which $1.19 \pm 0.12$ are predicted. Total visible and differential di-electron and di-muon photoproduction cross sections are extracted in a restricted phase space dominated by photon-photon collisions.