The ionizing fluxes from quasars and other active galactic nuclei (AGNs) are critical for interpreting their emission-line spectra and for photoionizing and heating the intergalactic medium. Using far-ultraviolet (FUV) spectra from the Cosmic Origins Spectrograph (COS) on the Hubble Space Telescope (HST), we directly measure the rest-frame ionizing continua and emission lines for 159 AGNs at redshifts 0.001<z_AGN_<1.476 and construct a composite spectrum from 475 to 1875 {AA}. We identify the underlying AGN continuum and strong extreme ultraviolet (EUV) emission lines from ions of oxygen, neon, and nitrogen after masking out absorption lines from the H I Ly{alpha} forest, 7 Lyman-limit systems (N_HI_>=10^17.2^/cm2) and 214 partial Lyman-limit systems (14.5<logN_HI_<17.2). The 159 AGNs exhibit a wide range of FUV/EUV spectral shapes, F_v_{propto}{nu}^{alpha}_{nu}_^, typically with -2<={alpha}_{nu}_<=0 and no discernible continuum edges at 912 {AA} (H I) or 504 {AA} (He I). The composite rest-frame continuum shows a gradual break at {lambda}_br_{approx} 1000 {AA}, with mean spectral index {alpha}_{nu}_=-0.83+/-0.09 in the FUV (1200-2000 {AA}) steepening to {alpha}_{nu}_=-1.41+/-0.15 in the EUV (500-1000 {AA}). We discuss the implications of the UV flux turnovers and lack of continuum edges for the structure of accretion disks, AGN mass inflow rates, and luminosities relative to Eddington values.