Investigation of methanol's surface chemistry on metals is a crucial step towards understanding the reactivity of this important chemical feedstock. Cu is a relevant metal for methanol synthesis and reforming, but due to the weak interaction of methanol with Cu, an atomic scale view of methanol's coverage-dependent ordering and self-assembly on Cu(111), the most abundant facet of most nanoparticles, has not yet been possible. Low and variable temperature scanning tunneling microscopy coupled with density functional theory reveal a coverage-dependent range of highly ordered structures stabilized by two hydrogen bonds per molecule. While extended chains that resemble the hydrogen-bonded zigzag structures reported for solid methanol are an efficient way to pack methanol at higher coverages, lower surface coverages yield isolated hexamer units. These hexamers form the same number of hydrogen bonds as the chains but appear to repel one another on the surface. Annealing treatments lead to the desorption of methanol with almost no decomposition. This data serves as a useful guide to both the preferred adsorption geometries and energies of a variety of methanol structures on Cu(111) surfaces as a function of surface coverage.