We present Large Binocular Telescope spectrophotometric observations of five low-redshift (z<0.070) compact star-forming galaxies (CSFGs) with extremely high emission-line ratios O32 = [OIII]5007/[OII]3727, ranging from 23 to 43. Galaxies with such high O32 are thought to be promising candidates for leaking large amounts of Lyman continuum (LyC) radiation and, at high redshifts, for contributing to the reionization of the Universe. The equivalent widths EW(Hbeta) of the Hbeta emission line in the studied galaxies are very high, ~350-520A, indicating very young ages for the star formation bursts, <3 Myr. All galaxies are characterized by low oxygen abundances 12+logO/H = 7.46 - 7.79 and low masses Mstar~10^6-10^7 Msun, much lower than the Mstar for known low-redshift LyC leaking galaxies, but probably more typical of the hypothetical population of low-luminosity dwarf LyC leakers at high redshifts. A broad Halpha emission line is detected in the spectra of all CSFGs, possibly related to expansion motions of supernova remnants. Such rapid ionized gas motions would facilitate the escape of the resonant Ly$alpha$ emission from the galaxy. We show that high O32 may not be a sufficient condition for LyC leakage and propose new diagnostics based on the HeI 3889/6678 and 7065/6678 emission-line flux ratios. Using these diagnostics we find that three CSFGs in our sample are likely to have density-bounded HII regions and are thus leaking large amounts of LyC radiation. The amount of leaking LyC radiation is probably much lower in the other two CSFGs.
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