  We test various conjectures about quantum gravity for six-dimensional string
compactifications in the framework of F-theory. Starting with a gauge theory
coupled to gravity, we analyze the limit in K\"ahler moduli space where the
gauge coupling tends to zero while gravity is kept dynamical. We show that such
a limit must be located at infinite distance in the moduli space. As expected,
the low-energy effective theory breaks down in this limit due to a tower of
charged particles becoming massless. These are the excitations of an
asymptotically tensionless string, which is shown to coincide with a critical
heterotic string compactified to six dimensions. For a more quantitative
analysis, we focus on a $U(1)$ gauge symmetry and use a chain of dualities and
mirror symmetry to determine the elliptic genus of the nearly tensionless
string, which is given in terms of certain meromorphic weak Jacobi forms. Their
modular properties in turn allow us to determine the mass-to-charge ratios of
certain string excitations near the tensionless limit. We then provide evidence
that the tower of asymptotically massless charged states satisfies the
(sub-)Lattice Weak Gravity Conjecture, the Completeness Conjecture, and the
Swampland Distance Conjecture. Quite remarkably, we find that the number
theoretic properties of the elliptic genus conspire with the balance of
gravitational and scalar forces of extremal black holes, such as to produce a
narrowly tuned charge spectrum of superextremal states. As a byproduct, we show
how to compute elliptic genera of both critical and non-critical strings, when
refined by Mordell-Weil $U(1)$ symmetries in F-theory.
