We report on a NuSTAR and XMM-Newton program that has observed a sample of three extremely luminous, heavily obscured WISE-selected active galactic nuclei (AGNs) at z ~ 2 across a broad X-ray band (0.1 − 79 keV). The parent sample, selected to be faint or undetected in the WISE 3.4 μm (W1) and 4.6 μm (W2) bands but bright at 12 μm (W3) and 22 μm (W4), are extremely rare, with only ~1000 so-called "W1W2-dropouts" across the extragalactic sky. Optical spectroscopy reveals typical redshifts of z ~ 2 for this population, implying rest-frame mid-IR luminosities of νL_ν(6 μm) ~ 6 × 10^(46) erg s^(−1) and bolometric luminosities that can exceed L_bol ~ 10^(14) L_☉. The corresponding intrinsic, unobscured hard X-ray luminosities are L(2–10 keV) ~ 4 × 10^(45) erg s^(−1) for typical quasar templates. These are among the most AGNs known, though the optical spectra rarely show evidence of a broad-line region and the selection criteria imply heavy obscuration even at rest-frame 1.5 μm. We designed our X-ray observations to obtain robust detections for gas column densities NH ≤ 10^(24) cm^(−2). In fact, the sources prove to be fainter than these predictions. Two of the sources were observed by both NuSTAR and XMM-Newton, with neither being detected by NuSTAR (f_(3–24 keV) ≲ 10^(−13) erg cm^(−2) s^(−1)), and one being faintly detected by XMM-Newton (f_(0.5–10 keV) ~ 5 × 10^(−15) erg cm^(−2) s^(−1)). A third source was observed only with XMM-Newton, yielding a faint detection (f_(0.5–10 keV) ~ 7 × 10^(−15) erg cm^(−2) s^(−1)). The X-ray data imply these sources are either X-ray weak, or are heavily obscured by column densities NH ≳ 10^(24) cm^(−2). The combined X-ray and mid-IR analysis seems to favor this second possibility, implying the sources are extremely obscured, consistent with Compton-thick, luminous quasars. The discovery of a significant population of heavily obscured, extremely luminous AGNs would not conform to the standard paradigm of a receding torus, in which more luminous quasars are less likely to be obscured, and instead suggests that an additional source of obscuration is present in these extreme sources.