Growth of Ag thin films on the twofold surface of a decagonal Al-Cu-Co quasicrystal is characterized by scanning tunneling microscopy, at different temperatures, and for coverages ranging from submonolayer to 11 monolayers. From prior work, three types of clean surface terraces are known to exist. By correlation with a bulk structural model, the major difference between them lies in their transition-metal (TM) content, two being aluminum-rich (0 and 15 at. % TM) and one being TM-rich (40–50 at. % TM). The present article focuses on understanding the difference between Ag film morphologies on these terminations, in terms of their chemical content. Growth is found to be smoother on the TM-rich terraces and rougher on the Al-rich ones. The first Ag atomic layer is even pseudomorphic on the TM-rich terraces. Roughness variation with temperature shows that the equilibrium morphology is two dimensional for TM-rich terraces and three dimensional for Al-rich terraces. The explanation of different growth modes in terms of different terrace compositions is supported by calculations of the adhesion energy of a Ag slab with Ag, Al, Cu, and Co slabs, using density-functional theory. For the Al-rich terraces, the roughness variation with temperature also indicates reentrant growth, i.e., anomalously smooth growth at low temperature due to kinetic effects.