We report on the magnetic properties of individual Fe atoms deposited on MgO(100) thin films probed by x-ray magnetic circular dichroism and scanning tunneling spectroscopy. We show that the Fe atoms have strong perpendicular magnetic anisotropy with a zero-field splitting of 14.0 +/- 0.3 meV/atom. This is a factor of 10 larger than the interface anisotropy of epitaxial Fe layers on MgO and the largest value reported for Fe atoms adsorbed on surfaces. The interplay between the ligand field at the O adsorption sites and spin-orbit coupling is analyzed by density functional theory and multiplet calculations, providing a comprehensive model of the magnetic properties of Fe atoms in a low-symmetry bonding environment.