  Detailed magnetization, specific heat, and $^7$Li nuclear magnetic resonance
(NMR) measurements on single crystals of the hyperhoneycomb Kitaev magnet
$\beta$-Li$_2$IrO$_3$ are reported. At high temperatures, {\cred anisotropy of
the magnetization is reflected by the different Curie-Weiss temperatures for
different field directions}, in agreement with the combination of a
ferromagnetic Kitaev interaction ($K$) and a negative off-diagonal anisotropy
($\Gamma$) as two leading terms in the spin Hamiltonian. At low temperatures,
magnetic fields applied along $a$ or $c$ have only a weak effect on the system
and reduce the N\'eel temperature from 38 K at 0 T to about 35.5 K at 14 T,
with no field-induced transitions observed up to 58 T on a powder sample. In
contrast, the field applied along $b$ causes a drastic reduction in the $T_N$
that vanishes around $H_c=2.8$ T giving way to a crossover toward a quantum
paramagnetic state. $^7$Li NMR measurements in this field-induced state reveal
a gradual line broadening and a continuous evolution of the line shift with
temperature, suggesting the development of local magnetic fields. The
spin-lattice relaxation rate shows a peak around the crossover temperature 40 K
and follows power-law behavior below this temperature.
