We present ab initio calculations for orbital moments and anisotropy energies of 3d and 5d adatoms on the Ag(001) surface, based on density functional theory, including Brooks' orbital polarization (OP) term, and applying a fully relativistic Korringa-Kohn-Rostoker-Green's function method. In general, we find unusually large orbital moments and anisotropy energies, e.g., in the 3d series, 2.57 mu(B) and +74 meV for Co, and, in the 5d series, 1.78 mu(B) and +42 meV for Os. These magnetic properties are determined mainly by the OP and even exist without spin-orbit coupling.