Hydrogen chemistry at extreme pressures is currently subject to extensive research due to the observed and predicted enhanced physical properties when hydrogen is incorporated in numerous binary systems. Despite the high reactivity of hydrogen, the noble metals (Cu, Ag, and Au) display an outstanding resilience to hydride formation, with no reports of a stable compound with a hydrogen molar ratio > 1 at room temperature. Here, through extreme compression and in situ laser heating of pure copper in a hydrogen atmosphere, we explore the affinity of these elements to adopt binary compounds. We report on the phase behavior and stabilities in the Cu–H system, analyzed via synchrotron X-ray diffraction, up to pressures of 50 GPa. We confirm the existence of the previously reported γ₀-CuH_0.15, γ₁-CuH_0.5, and ε-Cu₂H phases. Most notably, we report the highest hydrogen-content noble-metal hydride stable at room temperature to date: γ₂-CuH_0.65, which was synthesized through laser heating. This study furthers our understanding of hydrogen-transition metal chemistry and may find applicability in future hydrogen-storage applications.