We describe a possible new class of X-ray sources that have robust detections in ultra-deep Chandra data, yet have no detections at all in our deep multi-band GOODS Hubble Space Telescope (HST) ACS images, which represent the highest quality optical imaging obtained to date on these fields. These extreme X-ray / Optical ratio sources (EXOs) have values of Fx/Fopt at least an order of magnitude above those generally found for other AGN, even those that are harbored by reddened hosts. We thus infer two possible scenarios: (1) if these sources lie at redshifts z < 6, then their hosts need to be exceedingly underluminous, or more reddened, compared with other known sources; (2) if these sources lie above z ~ 6-7, such that even their Lyman-alpha emission is redshifted out of the bandpass of our ACS z(850) filter, then their optical and X-ray fluxes can be accounted for in terms of relatively normal L* hosts and moderate-luminosity AGN.