Let A be a DG Hopf algebra. The Gerstenhaber-Schack complex on A is a tricomplex whose 2-cocycles have infinitely many components, not just two. Thus "infinitesimal deformations" approximate infinitely many operations, two of which are a multiplication and a comultiplication. Although the associated deformation theory is undeveloped, this motivates the notion of an "A_\infty Hopf algebra." Roughly speaking, an A_\infty Hopf algebra is an R-module A equipped with structurally compatible operations \omega^{i,j}:A^i --> A^j, i,j >= 1, such that {A,\omega^{i,1}} is an A_\infty algebra and {A,\omega^{1,j}} is an A_\infty coalgebra. Structural compatibility requires a diagonal on the permutahedra, tensor products of A_\infty (co)algebras and (co)derivations with respect to a family of maps on a polytope. We mention several naturally occurring examples.