Room-temperature superconductivity has been a long-held dream and an area of intensive research. Recent experimental findings of superconductivity at 200 K in highly compressed hydrogen (H) sulfides have demonstrated the potential for achieving room-temperature superconductivity in compressed H-rich materials. We report first-principles structure searches for stable H-rich clathrate structures in rare earth hydrides at high pressures. The peculiarity of these structures lies in the emergence of unusual H cages with stoichiometries H_{24}, H_{29}, and H_{32}, in which H atoms are weakly covalently bonded to one another, with rare earth atoms occupying the centers of the cages. We have found that high-temperature superconductivity is closely associated with H clathrate structures, with large H-derived electronic densities of states at the Fermi level and strong electron-phonon coupling related to the stretching and rocking motions of H atoms within the cages. Strikingly, a yttrium (Y) H_{32} clathrate structure of stoichiometry YH_{10} is predicted to be a potential room-temperature superconductor with an estimated T_{c} of up to 303 K at 400 GPa, as derived by direct solution of the Eliashberg equation.