Gas hydrates are systems of prime importance. In particular, hydrogen hydrates are potential materials of icy satellites and comets, and may be used for hydrogen storage. We explore the H₂O-H₂ system at pressures in the range 0-100 GPa with ab initio variable-composition evolutionary simulations. According to our calculation and previous experiments, the H₂O-H₂ system undergoes a series of transformations with pressure, and adopts the known open-network clathrate structures (sII, C₀), dense "filled ice" structures (C₁, C₂) and two novel hydrate phases. One of these is based on the hexagonal ice framework and has the same H₂O:H₂ ratio (2:1) as the C₀ phase at low pressures and similar enthalpy (we name this phase Ih-C₀). The other newly predicted hydrate phase has a 1:2 H₂O:H₂ ratio and structure based on cubic ice. This phase (which we name C₃) is predicted to be thermodynamically stable above 38 GPa when including van der Waals interactions and zero-point vibrational energy, and explains previously mysterious experimental X-ray diffraction and Raman measurements. This is the hydrogen-richest hydrate and this phase has a remarkable gravimetric density (18 wt.%) of easily extractable hydrogen.