  Recently, the first JWST measurement of thermal emission from a rocky
exoplanet was reported. The inferred dayside brightness temperature of
TRAPPIST-1 b at 15 $\mu$m is consistent with the planet having no atmosphere
and therefore no mechanism by which to circulate heat to its nightside. In this
Letter, we compare the measured secondary eclipse depth of TRAPPIST-1 b to
predictions from a suite of self-consistent radiative-convective equilibrium
models in order to quantify the maximum atmospheric thickness consistent with
the observation. We find that plausible atmospheres (i.e., those that contain
at least 100 ppm CO$_2$) with surface pressures greater than 0.01 bar (0.1 bar)
are ruled out at 1$\sigma$ (3$\sigma$), regardless of the choice of background
atmosphere. Thicker atmospheres of up to 10 bar (100 bar) at 1$\sigma$
(3$\sigma$) are only allowed if the atmosphere lacks any strong absorbers
across the mid-IR wavelength range, a scenario that we deem unlikely. We
additionally model the emission spectra for bare-rock planets of various
compositions. We find that a variety of silicate surfaces match the measured
eclipse depth to within 1$\sigma$, and the best-fit grey albedo is $0.02 \pm
0.11$. We conclude that planned secondary eclipse observations at 12.8 $\mu$m
will serve to validate the high observed brightness temperature of TRAPPIST-1
b, but are unlikely to further distinguish among the consistent atmospheric and
bare-rock scenarios.
