We study the small population of high-redshift (z_em_>2.7) quasars detected by the Galaxy Evolution Explorer(GALEX), whose far-UV emission is not extinguished by intervening HI Lyman limit systems. We correlate almost all verified z_em_>2.7 quasars to the GALEX GR4 source catalog covering ~25000deg^2^, yielding 304 sources detected at signal-to-noise ratio (S/N)>3. However, ~50% of these are only detected in the GALEX NUV band, signaling the truncation of the FUV flux by low-redshift optically thick Lyman limit systems. We regard 52 quasars detected at S/N>3 to be most promising for Hubble Space Telescope follow-up, with an additional 114 quasars if we consider S/N>2 detections in the FUV. Combining the statistical properties of HI absorbers with the Sloan Digital Sky Survey (SDSS) quasar luminosity function, we predict a large all-sky population of ~200 quasars with z_em_>2.7 and i<~19 that should be detectable at the HeII edge at m_304_<21. However, SDSS provides just half of the NUV-bright quasars that should have been detected by SDSS and GALEX. With mock quasar photometry we revise the SDSS quasar selection function, finding that SDSS systematically misses quasars with blue u-g<~2 colors at 3<~z_em_3 and (FUV-NUV)2))