Connected finite graphs have unique prime factorizations with respect to the hierarchical product, but this property does not extend to connected infinite graphs. Here we show that all connected infinite rayless graphs have unique prime factorization with respect to the hierarchical product, and provide conditions under which this is also true for locally finite infinite graphs. Furthermore, we investigate the factorization properties of special classes of graphs, the structure of the automorphism groups of hierarchical products, and formulate several conjectures.