  The nonuniform black strings branch, which emerges from the critical
Gregory-Laflamme string, is numerically constructed in dimensions 6 <= D <= 11
and extended into the strongly non-linear regime. All the solutions are more
massive and less entropic than the marginal string. We find the asymptotic
values of the mass, the entropy and other physical variables in the limit of
large horizon deformations. By explicit metric comparison we verify that the
local geometry around the ``waist'' of our most nonuniform solutions is
cone-like with less than 10% deviation. We find evidence that in this regime
the characteristic length scale has a power-law dependence on a parameter along
the branch of the solutions, and estimate the critical exponent.
