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

Growth of homotopy groups of spheres via the Goodwillie-EHP Sequence