The geometries and energies of adsorption of up to five layers of water on the {111}, {110} and {100} surfaces of stoichiometric UO2 and PuO2 are studied computationally with Hubbard U-corrected density functional theory within the periodic boundary condition framework. The work builds on our recent study of the surface-bound water monolayers [J. Phys. Chem. C, 121, 1675 (2017)], and the water geometries within this first layer are used as the starting point for the present calculations. Significant variations are found in the per-layer adsorption energies, as a result of differing extents of intra-and inter-layer hydrogen bonding. After the adsorption of several additional layers, the effect of the surface-bound water geometries diminishes, and the average adsorption energy per water molecule is c. 0.5-0.6 eV (similar to that in bulk water) irrespective of the surface.