  Fusion data for $^{16}$O+$^{16}$O are analyzed by coupled-channels
calculations. It is shown that the calculated cross sections are sensitive to
the couplings to the $2^+$ and $3^-$ excitation channels even at low energies,
where these channels are closed. The sensitivity to the ion-ion potential is
investigated by applying a conventional Woods-Saxon potential and the
M3Y+repulsion potential, consisting of the M3Y double-folding potential and a
repulsive term that simulates the effect of the nuclear incompressibility. The
best overall fit to the data is obtained with a M3Y+repulsion potential which
produces a shallow potential in the entrance channel. The stepwise increase in
measured fusion cross sections at high energies is also consistent with such a
shallow potential. The steps are correlated with overcoming the barriers for
the angular momenta $L$ = 12, 14, 16, and 18. To improve the fit to the
low-energy data requires a shallower potential and this causes a even stronger
hindrance of fusion at low energies. It is therefore difficult, based on the
existing fusion data, to make an accurate extrapolation to energies that are of
interest to astrophysics.
