We present 20 WISE-selected galaxies with bolometric luminosities L_bol > 10^14 L_sun, including five with infrared luminosities L_IR = L(rest 8-1000 micron) > 10^14 L_sun. These "extremely luminous infrared galaxies," or ELIRGs, were discovered using the "W1W2-dropout" selection criteria which requires marginal or non-detections at 3.4 and 4.6 micron (W1 and W2, respectively) but strong detections at 12 and 22 micron in the WISE survey. Their spectral energy distributions are dominated by emission at rest-frame 4-10 micron, suggesting that hot dust with T_d ~ 450K is responsible for the high luminosities. These galaxies are likely powered by highly obscured AGNs, and there is no evidence suggesting these systems are beamed or lensed. We compare this WISE-selected sample with 116 optically selected quasars that reach the same L_bol level, corresponding to the most luminous unobscured quasars in the literature. We find that the rest-frame 5.8 and 7.8 micron luminosities of the WISE-selected ELIRGs can be 30-80% higher than that of the unobscured quasars. The existence of AGNs with L_bol > 10^14 L_sun at z > 3 suggests that these supermassive black holes are born with large mass, or have very rapid mass assembly. For black hole seed masses ~ 10^3 M_sun, either sustained super-Eddington accretion is needed, or the radiative efficiency must be