We have investigated by scanning tunneling microscopy the growth of Bi and Ag thin films on the fivefold surface of Al₆₃Cu₂₄Fe₁₃ and Al₇₂Pd_19.5Mn_8.5 quasicrystal, respectively. For both systems, we observe the formation of islands with magic height, corresponding to the stacking of a specific number of atomic layers. We interpret this unusual growth morphology in terms of quantum size effects, arising from the confinement of the electron within the film. The magic island heights are thus a direct manifestation of the electronic structure of the quasicrystalline substrates.