Using ab initio evolutionary simulations, we explore the entire range of possible stoichiometries for the Mg-O system at pressures of up to 850 GPa. In addition to MgO, our calculations find that two extraordinary compounds MgO2 and Mg3O2 become thermodynamically stable at 116 GPa and 500 GPa, respectively. Detailed chemical bonding analysis shows large charge transfer in all magnesium oxides. MgO2 contains peroxide ions [O-O](2-), while non-nuclear electron density maxima play the role of anions in the electride compound Mg3O2. The latter compound is calculated to have a much narrower band gap compared to MgO and MgO2. We discuss the conditions under which MgO2 and Mg3O2 could exist in planetary interiors.