Despite the unique X-ray behavior of the compact bursting X-ray source MXB 1730-335, the "Rapid Burster" (RB) in the highly reddened globular cluster Liller 1, to date there has been no known optical/IR counterpart for the object, no precise astrometric solution that correlates X-ray, radio, and optical positions and thus restricts the number of possible candidates, nor even published IR images of the field. We solve a previous radio/X-ray positional discrepancy, presenting the results of precise Chandra X-ray imaging, which definitively show that the radio source is positionally aligned with MXB 1730-335. At the same time, we have detected three additional low-luminosity (L_X ~ 10³⁴ ergs s^-1) X-ray sources within two core radii, which are possibly quiescent low-mass X-ray binaries. We present both ground-based and Hubble Space Telescope (HST) infrared imaging of the field (in quiescent and bursting X-ray states of the RB), together with the necessary astrometric solution to overlay the radio/X-ray source positions. Even at HST resolution, the RB field is very complex and there are multiple candidates. No object of unusual color or of substantial variability in quiescent versus active or burst versus nonburst states is identified. Further, more sensitive HST NICMOS and/or ground-based, adaptive optics observations are needed to confidently identify the proper counterpart. In the case of the RB, uncertain but plausible calculations on the effects of the burst on the binary companion indicate that detection of a variable candidate should be feasible.