We present the optical and infrared properties of 39 extremely radio- loud galaxies discovered by cross-matching the Subaru/Hyper Suprime-Cam (HSC) deep optical imaging survey and VLA/FIRST 1.4 GHz radio survey. The recent Subaru/HSC strategic survey revealed optically faint radio galaxies (RG) down to gAB ~26, opening a new parameter space of extremely radio-loud galaxies (ERGs) with radio- loudness parameter of logrest=log(f1.4GHz,rest∕fg,rest)4. Because of their optical faintness and small number density of ~1 deg-2, such ERGs were difficult to find in the previous wide but shallow or deep but small area optical surveys. ERGs show intriguing properties that are different from the conventional RGs: (1) most ERGs reside above or on the star-forming main- sequence and some of them might be low-mass galaxies with log(M⋆∕M⊙)10 . (2) ERGs exhibit a high specific black hole accretion rate, reaching the order of the Eddington limit. The intrinsic radio loudness ( int ), defined by the ratio of jet power over bolometric radiation luminosity, is one order of magnitude higher than that of radio quasars. This suggests that ERGs harbor a unique type of active galactic nuclei (AGN) that show both powerful radiations and jets. Therefore, ERGs are prominent candidates of very rapidly growing black holes reaching Eddington-limited accretion just before the onset of intensive AGN feedback.