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 $g_mathrm{AB} sim 26$, opening a new parameter space of extremely radio-loud galaxies (ERGs) with radio-loudness parameter of $log mathcal{R}_mathrm{rest} = log (f_{1.4 mathrm{GHz,rest}}/f_{g,mathrm{rest}}) >4$. Because of their optical faintness and small number density of $sim1~$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 $log (M_star/M_odot) < 10$. (2) ERGs exhibit a high specific black hole accretion rate, reaching the order of the Eddington limit. The intrinsic radio-loudness ($mathcal{R}_mathrm{int}$), 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.
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