  We study, from both a theoretical and observational perspective, the physical
origin and spectroscopic impact of extreme nebular emission in high-redshift
galaxies. The nebular continuum, which can appear during extreme starbursts, is
of particular importance as it tends to redden UV slopes and has a significant
contribution to the UV luminosities of galaxies. Furthermore, its shape can be
used to infer the gas density and temperature of the ISM. First, we provide a
theoretical background, showing how different stellar populations (SPS models,
IMFs, and stellar temperatures) and nebular conditions impact observed galaxy
spectra. We demonstrate that, for systems with strong nebular continuum
emission, 1) UV fluxes can increase by up to 0.7~magnitudes (or more in the
case of hot/massive stars) above the stellar continuum, which may help
reconcile the surprising abundance of bright high-redshift galaxies and the
elevated UV luminosity density at $z>10$, 2) at high gas densities, UV slopes
can redden from $\beta\lesssim-2.5$ to $\beta\sim-1$, 3) observational
measurements of $\xi_{ion}$ are grossly underestimated, and 4) UV downturns
from two-photon emission can masquerade as DLAs. Second, we present a dataset
of 58 galaxies observed with NIRSpec on JWST at $2.5<z<9.0$ that are selected
to have strong nebular continuum emission via the detection of the Balmer jump.
Five of the 58 spectra are consistent with being dominated by nebular emission,
exhibiting both a Balmer jump and a UV downturn consistent with two-photon
emission. For some galaxies, this may imply the presence of hot massive stars
and a top-heavy IMF. We conclude by exploring the properties of
spectroscopically confirmed $z>10$ galaxies, finding that UV slopes and UV
downturns are in some cases redder or steeper than expected from SPS models,
which may hint at more exotic (e.g. hotter/more massive stars or AGN) ionizing
sources.
