  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. 
Beating the 3 dB Limit for Intracavity Squeezing and Its Application to   Nondemolition Qubit Readout