  Geometrically non-Higgsable seven-branes carry gauge sectors that cannot be
broken by complex structure deformation, and there is growing evidence that
such configurations are typical in F-theory. We study strongly coupled physics
associated with these branes. Axiodilaton profiles are computed using
Ramanujan's theories of elliptic functions to alternative bases, showing
explicitly that the string coupling is order one in the vicinity of the brane;
that it sources nilpotent $SL(2,\mathbb{Z})$ monodromy and therefore the
associated brane charges are modular; and that essentially all F-theory
compactifications have regions with order one string coupling. It is shown that
non-perturbative $SU(3)$ and $SU(2)$ seven-branes are related to weakly coupled
counterparts with D7-branes via deformation-induced Hanany-Witten moves on
$(p,q)$ string junctions that turn them into fundamental open strings; only the
former may exist for generic complex structure. D3-brane near these and the
Kodaira type II seven-branes probe Argyres-Douglas theories. The BPS states of
slightly deformed theories are shown to be dyonic string junctions.
