  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
$E(B\!-\!V)_{\text{AGN}}$ 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 restframe 2-10 keV ($L_{\text{X}}$) and restframe AGN 6
$\mu{\text{m}}$ ($L_{\text{MIR}}$), 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 ($R_{L_{\text{X}}}$), we find a
significant fraction of sources with column densities approaching
$N_{\text{H}}>$ 10$^{\text{24}}$ cm$^{-{\text{2}}}$, suggesting that
multiwavelength observations are necessary to account for the population of
heavily obscured AGNs. We simulate the underlying $N_{\text{H}}$ 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.
