The development and local structure of height-selected 3-layer Ag islands on fivefold surfaces of icosahedral Al-Pd-Mn quasicrystals is characterized by STM for Ag deposition at 365 K. Heterogeneous nucleation of pseudomorphic single layer high islands is followed by rapid formation of 2nd and 3rd layers and subsequent lateral spreading, where each of these 3 layers consists of a family of nonfcc structures. The behavior is elucidated by step dynamics modeling incorporating strain buildup for larger islands, enhanced binding in higher layers, and height selection due to quantum size effects.