The Kreimer-Connes Hopf algebraic formulation of quantum field theoretic renormalization provides a powerful framework to study renormalized QFTs at high orders in perturbation theory. But QFTs also have non-perturbative features, conventionally identified with "instantons" or "renormalons". Using methods of resurgent asymptotics, we can extract non-perturbative information directly from a perturbative analysis, here illustrated starting from the perturbative Hopf algebraic formalism.