We have performed inelastic scanning tunneling spectroscopy (ISTS) on individual Fe atoms adsorbed on a Ag(111) surface. ISTS reveals a magnetization excitation with a lifetime of about 400 fs, which decreases linearly upon application of a magnetic field. Surprisingly, we find that the g factor, which characterizes the shift in energy of the excitation in a magnetic field, is g = 3.1 instead of the expected value of 2. This very large g shift can be understood when considering the complete electronic structure of both the Ag(111) surface state and the Fe atom, as shown by ab initio calculations of the magnetic susceptibility.