  We bound the volume of the homotopy groups of the 2-local Goodwillie
approximations of a sphere in terms of the amount of $2$-torsion in the stable
stems, providing a Goodwillie-theoretic refinement of a result of Burklund and
Senger (arXiv:2203.00670). At the $2^k$-excisive approximation, this bound is
obtained by `multiplying the stable answer by a polynomial of degree $k$'. The
main tool is Behrens' Goodwillie-EHP Long Exact Sequence.

Algebraic Topology