Novel properties of metal monolayers and superlattices are likely to depend on the detailed morphology of monolayer growth. We find, using scanning tunneling microscopy, that the "simple" system of Ag on room-temperature Au(111) exhibits a morphological instability in the formation of the first monolayer. Monolayer-high Ag protrusions grow outward from atomic steps onto atomic terraces. The structure of the deposits on a nanometer scale shows that Ag atoms diffuse freely on Au terrace but are not mobile along or across step edges that separate the terraces. The result is a complex growth front characteristic of diffusion-limited aggregation with local relaxation. This causes strong inhomogeneity in local coverage, so that even at 1.1 ml average coverage there are large areas [(1000 angstrom)2] of exposed Au. Experiments with step structures intentionally created by tip impact suggest that atoms on step edges are not highly mobile at room temperature, so that the dominant mechanism for local smoothing of these deposits is relaxation intrinsic to the aggregation process.