  We apply the coupled cluster method to high orders of approximation and exact
diagonalizations to study the ground-state properties of the triangular-lattice
spin-$s$ Heisenberg antiferromagnet. We calculate the fundamental ground-state
quantities, namely, the energy $e_0$, the sublattice magnetization $M_{\rm
sub}$, the in-plane spin stiffness $\rho_s$ and the in-plane magnetic
susceptibility $\chi$ for spin quantum numbers $s=1/2, 1, \ldots, s_{\rm max}$,
where $s_{\rm max}=9/2$ for $e_0$ and $M_{\rm sub}$, $s_{\rm max}=4$ for
$\rho_s$ and $s_{\rm max}=3$ for $\chi$. We use the data for $s \ge 3/2$ to
estimate the leading quantum corrections to the classical values of $e_0$,
$M_{\rm sub}$, $\rho_s$, and $\chi$. In addition, we study the magnetization
process, the width of the 1/3 plateau as well as the sublattice magnetizations
in the plateau state as a function of the spin quantum number $s$.
