  FRB 20210912A is a fast radio burst (FRB), detected and localised to
sub-arcsecond precision by the Australian Square Kilometre Array Pathfinder. No
host galaxy has been identified for this burst despite the high precision of
its localisation and deep optical and infrared follow-up, to 5-$\sigma$ limits
of $R=26.7$ mag and $K_\mathrm{s}=24.9$ mag with the Very Large Telescope. The
combination of precise radio localisation and deep optical imaging has almost
always resulted in the secure identification of a host galaxy, and this is the
first case in which the line-of-sight is not obscured by the Galactic disk. The
dispersion measure of this burst,
$\mathrm{DM_{FRB}}=1233.696\pm0.006~\mathrm{pc}\ \mathrm{cm}^{-3}$, allows for
a large source redshift of $z>1$ according to the Macquart relation. It could
thus be that the host galaxy is consistent with the known population of FRB
hosts, but is too distant to detect in our observations ($z>0.7$ for a host
like that of the first repeating FRB source, FRB 20121102A); that it is more
nearby with a significant excess in $\mathrm{DM_{host}}$, and thus dimmer than
any known FRB host; or, least likely, that the FRB is truly hostless. We
consider each possibility, making use of the population of known FRB hosts to
frame each scenario. The fact of the missing host has ramifications for the FRB
field: even with high-precision localisation and deep follow-up, some FRB hosts
may be difficult to detect, with more distant hosts being the less likely to be
found. This has implications for FRB cosmology, in which high-redshift
detections are valuable.
