  We provide an explicit expression for the strong magnetic field limit of the
Heisenberg-Euler effective Lagrangian for both scalar and spinor quantum
electrodynamics. To this end, we show that the strong magnetic field behavior
is fully determined by one-particle reducible contributions discovered only
recently. The latter can efficiently be constructed in an essentially algebraic
procedure from lower-order one-particle reducible diagrams. Remarkably, the
leading strong magnetic field behavior of the all-loop Heisenberg-Euler
effective Lagrangian only requires input from the one-loop Lagrangian. Our
result revises previous findings based exclusively on one-particle irreducible
contributions. In addition we briefly discuss the strong electric field limit
and comment on external field QED in the large $N$ limit.
