The isotropic magnetic moment of a free atom is shown to develop giant magnetic anisotropy energy due to symmetry reduction at an atomically ordered surface. Single cobalt atoms deposited onto platinum (111) are found to have a magnetic anisotropy energy of 9 millielectron volts per atom arising from the combination of unquenched orbital moments (1.1 Bohr magnetons) and strong spin-orbit coupling induced by the platinum substrate. By assembling cobalt nanoparticles containing up to 40 atoms, the magnetic anisotropy energy is further shown to be dependent on single-atom coordination changes. These results confirm theoretical predictions and are of fundamental value to understanding how magnetic anisotropy develops in finite-sized magnetic particles.