  While the squeezing of a propagating field can, in principle, be made
arbitrarily strong, the cavity-field squeezing is subject to the well-known 3
dB limit, and thus has limited applications. Here, we propose the use of a
fully quantum degenerate parametric amplifier (DPA) to beat this squeezing
limit. Specifically, we show that by {\it simply} applying a two-tone driving
to the signal mode, the pump mode can, {\it counterintuitively}, be driven by
the photon loss of the signal mode into a squeezed steady state with, in
principle, an {\it arbitrarily high} degree of squeezing. Furthermore, we
demonstrate that this intracavity squeezing can increase the signal-to-noise
ratio of longitudinal qubit readout {\it exponentially} with the degree of
squeezing. Correspondingly, an improvement of the measurement error by {\it
many orders of magnitude} can be achieved even for modest parameters. In stark
contrast, using intracavity squeezing of the semiclassical DPA {\it cannot}
practically increase the signal-to-noise ratio and thus improve the measurement
error. Our results extend the range of applications of DPAs and open up new
opportunities for modern quantum technologies.
