The surface morphology of single- and multilayer Ge films on GaAs(100) has been studied using low-energy electron diffraction and scanning tunneling microscopy (STM). After deposition of 0.5 monolayer, the Ge atoms form dimers with Ga atoms after annealing to 875 K, resulting in a (1X2) ordered surface layer. A Ge multilayer deposited at 700 K shows a mixture of (1X2) and (2x1) phases. When annealed at higher temperatures, the (2X1) structure begins to disappear, and, eventually, only the (1X2) phase remains after annealing at 875 K. STM images of the surface reveal faceted clusters of Ge that are surrounded by relatively smooth areas of (1X2) order. Based on these observations, we suggest that the Ge film grows on GaAs(100) by the Stranski-Krastanov mechanism at high temperatures. Because the lattice strain is negligible, the driving force for this growth mode is a reduction in the total surface energy by minimizing substrate-induced electronic effects. Comparison of Ge growth on GaAs(100), GaAs(110), and Si(100) substrates shows the important role of surface and bulk electronic structure in determining the growth mode. The observed antiphase-domain-free GaAs growth on Ge/GaAs(100) is explained based on our results.