Iron arsenide superconductors based on the material LaFeAsO(1-x)F(x) are characterized by a two-dimensional Fermi surface (FS) consisting of hole and electron pockets yielding structural and antiferromagnetic transitions at x=0. Electron doping by substituting O(2-) with F(-) suppresses these transitions and gives rise to superconductivity with a maximum T(c) of 26 K at x=0.1. However, the over-doped region cannot be accessed due to the poor solubility of F(-) above x=0.2. Here we overcome this problem by doping LaFeAsO with hydrogen. We report the phase diagram of LaFeAsO(1-x)H(x) (x<0.53) and, in addition to the conventional superconducting dome seen in LaFeAsO(1-x)F(x), we find a second dome in the range 0.21<x<0.53, with a maximum T(c) of 36 K at x=0.3. Density functional theory calculations reveal that the three Fe 3d bands (xy, yz and zx) become degenerate at x=0.36, whereas the FS nesting is weakened monotonically with x. These results imply that the band degeneracy has an important role to induce high T(c).