  This paper presents a comprehensive analysis of four triply eclipsing triple
star systems, namely TIC 88206187, TIC 14839347, TIC 298714297, and TIC
66893949. The four systems with third-body eclipses were found in the TESS
lightcurves from among a sample of ~400 matches between known eclipsing
binaries and the Gaia DR3 Non-Single Star (NSS; Gaia Collaboration 2022;
Pourbaix et al. 2022) solution database. We combined photometric lightcurves,
eclipse timing variations, archival spectral energy distributions, and
theoretical evolution tracks in a robust photodynamical analysis to determine
the orbital and system parameters. The triples have outer periods of 52.9,
85.5, 117, and 471 days, respectively. All dozen stars have masses $\lesssim$
2.6 M$_\odot$. The systems are quite flat with mutual inclination angles
between the inner and outer orbital planes that are all $\lesssim 4^\circ$. The
outer mass ratios range from 0.39--0.76, consistent with our earlier collection
of compact triply eclipsing triples. TIC 88206187 exhibits a fractional radius
of the outer tertiary component $(r_B \equiv R_B/a_{\rm out})$ exceeding 0.1
(only the third such system known), and we consider its future evolution.
Finally, we compare our photodynamical analysis results and the orbital
parameters given in the Gaia DR3 NSS solutions, indicating decent agreement,
but with the photodynamical results being more accurate.
