  We advocate the study of external-field quantum electrodynamics with $N$
charged particle flavors. Our main focus is on the Heisenberg-Euler effective
action for this theory in the large $N$ limit which receives contributions from
all loop orders. The contributions beyond one loop stem from one-particle
reducible diagrams. We show that specifically in constant electromagnetic
fields the latter are generated by the one-loop Heisenberg-Euler effective
Lagrangian. Hence, in this case the large $N$ Heisenberg-Euler effective action
can be determined explicitly at any desired loop order. We demonstrate that
further analytical insights are possible for electric-and magnetic-like field
configurations characterized by the vanishing of one of the secular invariants
of the electromagnetic field and work out the all-orders strong field limit of
the theory.
