[Abridged] We have carried out a deep X-ray and optical survey with Chandra and HST of low-extinction regions in the Galactic bulge. Here we present the results of a search for low-luminosity (L_X <~ 1e34 ergs/s) accreting binaries in the region closest to the Galactic Center, at an angular offset of 1.4deg, that we have named the Limiting Window. Based on their blue optical colors, excess Halpha fluxes, and high X-ray--to--optical flux ratios, we identify three likely cataclysmic variables (CVs). Distance estimates put these systems farther than >~2 kpc. Based on their Halpha-excess fluxes and/or high X-ray--to--optical flux ratios, we find 22 candidate accreting binaries; however, the properties of some can also be explained if they are dMe stars or active galaxies. We investigate the CV number density towards the bulge and find that the number of observed candidate CVs is consistent with or lower than the number expected for a constant CV-to-star ratio that is fixed to the local value. The X-ray properties of two likely CVs are similar to those of the faint, hard X-ray sources in the Galactic-Center region that have been explained by (mainly) magnetic CVs. If our candidates belong to the same population, they would be the first members to be optically identified; optical or infrared identification of their Galactic-Center analogs would be impossible due to the higher obscuration. We speculate that all Galactic hard X-ray sources in our field can be explained by magnetic CVs.
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