  Novel sequences of approximants to solutions of Painlev\'e II on finite
intervals of the real line, with Neumann boundary conditions, are constructed.
Numerical experiments strongly suggest convergence of these sequences in a
surprisingly wide range of cases, even ones where ordinary perturbation series
fail to converge. These sequences are here labeled extraordinary because of
their unusual properties. Each element of such a sequence is defined on its own
interval. As the sequence (apparently) converges to a solution of the
corresponding boundary value problem for Painlev\'e II, these intervals
themselves (apparently) converge to the defining interval for that problem, and
an associated sequence of constants (apparently) converges to the constant term
in the Painlev\'e II equation itself. Each extraordinary sequence is
constructed in a nonlinear fashion from a perturbation series approximation to
the solution of a supplementary boundary value problem, involving a
generalization of Painlev\'e II that arises in studies of electrodiffusion.
