Quasicrystals exhibit exotic properties inherited from the self-similarity of their long-range ordered, yet aperiodic, structure. The recent realization of an optical quasicrystal lattice for ultracold atoms paves the way to the study of correlated bosons in such structures, but the regime of strong interactions remains largely unexplored. Here, we determine the quantum phase diagram of two-dimensional correlated bosons in a quasicrystal potential. Using large-scale Monte Carlo calculations, we find evidence of a superfluid-to-Bose glass transition, and determine the critical line. Moreover, we show that strong interactions can stabilize a Mott insulator phase, possibly with spontaneously broken eightfold symmetry. Our results drive prospects to experimental investigation and, in particular, to the observation of the still elusive Bose glass phase in two dimensions.