  For certain quantum field theories, the Kreimer-Connes Hopf-algebraic
approach to renormalization reduces the Dyson-Schwinger equations to a system
of non-linear ordinary differential equations for the expansion coefficients of
the renormalized Green's function. We apply resurgent asymptotic analysis to
find the trans-series solutions which provide the non-perturbative completion
of these formal Dyson-Schwinger expansions. We illustrate the general approach
with the concrete example of four dimensional massless Yukawa theory,
connecting with the exact functional solution found by Broadhurst and Kreimer.
The trans-series solution is associated with the iterative form of the
Dyson-Schwinger equations, and displays renormalon-like structure of
integer-repeated Borel singularities. Extraction of the Stokes constant is
possible due to a property we call `functional resurgence'.
