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It is shown that low-collisionality plasmas cannot support linearly polarizedshear-Alfv\'en fluctuations above a critical amplitude $\delta B_{\perp}/B_{0}\sim \beta^{\,-1/2}$, where $\beta$ is the ratio of thermal to magneticpressure. Above this cutoff, a developing fluctuation will generate a pressureanisotropy that is sufficient to destabilize itself through the parallelfirehose instability. This causes the wave frequency to approach zero,interrupting the fluctuation before any oscillation. The magnetic field linesrapidly relax into a sequence of angular zig-zag structures. Such a restrictivebound on shear-Alfv\'en-wave amplitudes has far-reaching implications for thephysics of magnetized turbulence in the high-$\beta$ conditions prevalent inmany astrophysical plasmas, as well as for the solar wind at $\sim 1\mathrm{AU}$ where $\beta \gtrsim 1$.

A stringent limit on the amplitude of Alfv\'enic perturbations in high-beta low-collisionality plasmas