We present a large sample of infrared-luminous candidate active galactic nuclei (AGNs) that lack X-ray detections in Chandra, XMM-Newton, and NuSTAR fields. We selected all optically detected SDSS sources with redshift measurements, combined additional broadband photometry from WISE, UKIDSS, 2MASS, and GALEX, and modeled the spectral energy distributions (SEDs) of our sample sources. We parameterize nuclear obscuration in our SEDs with E(B-V)_AGN_ and uncover thousands of powerful obscured AGNs that lack X-ray counterparts, many of which are identified as AGN candidates based on straightforward WISE photometric criteria. Using the observed luminosity correlation between rest-frame 2-10 keV (L_X_) and rest-frame AGN 6um (L_MIR_), we estimate the intrinsic X-ray luminosities of our sample sources and combine these data with flux limits from X-ray catalogs to determine lower limits on nuclear obscuration. Using the ratio of intrinsic-to-observed X-ray luminosity (R_L_X__), we find a significant fraction of sources with column densities approaching N_H_>10^24^/cm^2^, suggesting that multiwavelength observations are necessary to account for the population of heavily obscured AGNs. We simulate the underlying N_H_ distribution for the X-ray non-detected sources in our sample through survival analysis, and confirm the presence of AGN activity via X-ray stacking. Our results point to a considerable population of extremely obscured AGNs undetected by current X-ray observatories.