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In this work we report the discovery and analysis of six new compact triply eclipsing triple star systems found with the TESS mission: TICs 37743815, 42565581, 54060695, 178010808, 242132789, and 456194776. All of these exhibit distinct third body eclipses where the inner eclipsing binary (EB) occults the third (`tertiary') star, or vice versa. We utilized the TESS photometry, archival photometric data, and available archival spectral energy distribution curves (SED) to solve for the properties of all three stars, as well as many of the orbital elements. We describe in detail our SED fits, search of the archival data for the outer orbital period, and the final global photodynamical analyses. From these analyses we find that all six systems are coplanar to within $0^\circ$ - $5^\circ$, and are viewed nearly edge on (i.e., within a couple of degrees). The outer orbital periods and eccentricities of the six systems are {$P_{\rm out}$ (days), $e$}: {68.7, 0.36}, {123, 0.16}, {60.7, 0.01}, {69.0, 0.29}, {41.5, 0.01}, {93.9, 0.29}, respectively, in the order the sources are listed above. The masses of all 12 EB stars were in the range of 0.7-1.8 M$_\odot$ and were situated near the main sequence. By contrast, the masses and radii of the tertiary stars ranged from 1.5-2.3 M$_\odot$ and 2.9-12 R$_\odot$, respectively. We use this information to estimate the occurrence rate of compact flat triple systems.