We report on the properties of the most massive ultra-compact dwarf galaxy ( UCD ) in the nearby Virgo Cluster of galaxies using imaging from the Next Generation Virgo Cluster Survey ( NGVS ) and spectroscopy from Keck/DEIMOS . This object ( M59-UCD3 ) appears to be associated with the massive Virgo galaxy M59 ( NGC 4621 ) , has an integrated velocity dispersion of 78 { km s ^ { -1 } } , a dynamical mass of 3.7 \times 10 ^ { 8 } M _ { \odot } , and an effective radius ( R _ { e } ) of 25 pc . With an effective surface mass density of 9.4 \times 10 ^ { 10 } M _ { \odot } { kpc } ^ { -2 } , it is the densest galaxy in the local Universe discovered to date , surpassing the density of the luminous Virgo UCD , M60-UCD1 . M59-UCD3 has a total luminosity of M _ { g ^ { \prime } } = -14.2 mag , and a spectral energy distribution consistent with an old ( 14 Gyr ) stellar population with [ Fe/H ] = 0.0 and [ \alpha / Fe ] = +0.2 . We also examine deep imaging around M59 and find a broad low surface brightness stream pointing towards M59-UCD3 , which may represent a tidal remnant of the UCD progenitor . This UCD , along with similar objects like M60-UCD1 and M59cO , likely represents an extreme population of tidally stripped galaxies more akin to larger and more massive compact early-type galaxies than to nuclear star clusters in present-day dwarf galaxies .