Room temperature adsorption of molecular iodine on Ag(111) has been studied by scanning tunneling microscopy (STM), low energy electron diffraction, Auger electron spectroscopy with factor analysis, and density functional theory (DFT). At the chemisorption stage, iodine first forms a (3×3)R30° structure. Further iodine dosing leads to continuous commensurate-incommensurate phase transition, taking place via the formation of striped superheavy domain walls. As a result, the uniaxially compressed (13 ×3-R30°) phase is formed at an iodine coverage (θ) of 0.38 ML. At θ > 0.38 ML, first-order phase transition begins, leading to the formation of hexagonal moiré-like phases, which exhibit an anomalously large corrugation in STM (0.8-2.3 Å). In the range of 0.40-0.43 ML, the compression of hexagonal phases occurs, which ends at the formation of the (7 × 7)R21.8° structure at saturation. The DFT calculations allow us to explain the anomalous atomic corrugation of the hexagonal phases by the strong violation of the atomic structure of the substrate including up to ten layers of silver. Iodine dosing above 0.43 ML leads to the growth of 2D islands of silver iodide. The STM images of the silver iodide surface demonstrate a clear visible hexagonal superstructure with a periodicity of 25 Å superimposed with a quasi-hexagonal atomic modulation. DFT calculations of the atomic structure of AgI islands point to the formation of a sandwich-like double layer honeycomb structure similar to the case of I/Ag(100).