  Frustrated spin-1/2 chains, despite the apparent simplicity, exhibit
remarkably rich phase diagram comprising vector-chiral (VC), spin-density-wave
(SDW) and multipolar/spin-nematic phases as a function of the magnetic field.
Here we report a study of $\beta$-TeVO$_4$, an archetype of such compounds,
based on magnetization and neutron diffraction measurements up to 25 T. We find
the transition from the helical VC ground state to the SDW state at $\sim$3 T
for the magnetic field along the $a$ and $c$ crystal axes, and at $\sim$9 T for
the field along the $b$ axis. The high-field (HF) state, existing above
$\sim$18 T, i.e., above $\sim$1/2 of the saturated magnetization, is an
incommensurate magnetically ordered state and not the spin-nematic state, as
theoretically predicted for the isotropic frustrated spin-1/2 chain. The HF
state is likely driven by sizable interchain interactions and symmetric
intrachain anisotropies uncovered in previous studies. Consequently, the
potential existence of the spin nematic phase in $\beta$-TeVO$_4$ is limited to
a narrow field range, i.e., a few tenths of a tesla bellow the saturation of
the magnetization, as also found in other frustrated spin-1/2 chain compounds.
