Community access to deep (i ~ 25), highly-multiplexed optical and near-infrared multi-object spectroscopy (MOS) on 8-40m telescopes would greatly improve measurements of cosmological parameters from LSST. The largest gain would come from improvements to LSST photometric redshifts, which are employed directly or indirectly for every major LSST cosmological probe; deep spectroscopic datasets will enable reduced uncertainties in the redshifts of individual objects via optimized training. Such spectroscopy will also determine the relationship of galaxy SEDs to their environments, key observables for studies of galaxy evolution. The resulting data will also constrain the impact of blending on photo-z's. Focused spectroscopic campaigns can also improve weak lensing cosmology by constraining the intrinsic alignments between the orientations of galaxies. Galaxy cluster studies can be enhanced by measuring motions of galaxies in and around clusters and by testing photo-z performance in regions of high density. Photometric redshift and intrinsic alignment studies are best-suited to instruments on large-aperture telescopes with wider fields of view (e.g., Subaru/PFS, MSE, or GMT/MANIFEST) but cluster investigations can be pursued with smaller-field instruments (e.g., Gemini/GMOS, Keck/DEIMOS, or TMT/WFOS), so deep MOS work can be distributed amongst a variety of telescopes. However, community access to large amounts of nights for surveys will still be needed to accomplish this work. In two companion white papers we present gains from shallower, wide-area MOS and from single-target imaging and spectroscopy.