  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.
