  We study the ground state properties of antiferromagnetic $J_1$-$J_2$ chains
with half-integer spins ranging from $S=\frac{3}{2}$ to $S=\frac{11}{2}$ using
the density-matrix renormalization group method. We map out the ground-state
phase diagrams as a function of $\frac{J_2}{J_1}$ containing topological phases
with alternating $\frac{2S-1}{2}$ and $\frac{2S+1}{2}$ valence bonds. We
identify these topological phases and their boundaries by calculating the
string order parameter, the dimer order parameter, and the spin gap for those
high-$S$ systems in thermodynamic limit (finite size scaling). We find that
these topological regions narrow down inversely with $S$ and converge to a
single point at $\frac{J_2}{J_1}=\frac{1}{4}$ in the classical limit -- a
critical threshold between commensurate and incommensurate orders. In addition,
we extend the discussion of the Majumder-Ghosh state, previously noted only for
$S=\frac{1}{2}$, and speculate its possible presence as a ground state in
half-integer high spin systems over a substantial range of $\frac{J_2}{J_1}$
values.
