  We show that the one-loop Euler-Heisenberg QED effective Lagrangian in a
constant background field acquires a very different non-perturbative
trans-series structure at two-loop and higher-loop order in the fine structure
constant. Beyond one-loop, virtual particles interact, causing fluctuations
about the instantons, whereby the simple poles of the one-loop Borel transform
become branch points. We illustrate this in detail at two-loop order using
Ritus's seminal result for the renormalized two-loop effective Lagrangian as an
exact double-integral representation, and propose a possible new approach to
computations at higher loop order. Our methods yield remarkably accurate
extrapolations from weak-field to strong-field, and from magnetic to electric
background field, at both one-loop and two-loop order, based on surprisingly
little perturbative input.
