  We study the Heisenberg-Euler effective action in constant electromagnetic
fields $\bar{F}$ for QED with $N$ charged particle flavors of the same mass and
charge $e$ in the large $N$ limit characterized by sending $N\to\infty$ while
keeping $Ne^2\sim e\bar{F}\sim N^0$ fixed. This immediately implies that
contributions that scale with inverse powers of $N$ can be neglected and the
resulting effective action scales linearly with $N$. Interestingly, due to the
presence of one-particle reducible diagrams, even in this limit the
Heisenberg-Euler effective action receives contributions of arbitrary loop
order. In particular for the special cases of electric- and magnetic-like field
configurations we construct an explicit expression for the associated effective
Lagrangian that, upon extremization for two constant scalar coefficients,
allows to evaluate its full, all-order result at arbitrarily large field
strengths. We demonstrate that our manifestly nonperturbative expression
correctly reproduces the known results for the Heisenberg-Euler effective
action at large $N$, namely its all-loop strong field limit and its low-order
perturbative expansion in powers of the fine-structure constant.
