In the last pages of "Equations Différentielles à points singuliers réguliers", Deligne provides a proof, attributed to Brieskorn, of the so-called local monodromy theorem (on the quasi-unipotence of the monodromy operator acting on the cohomology of a degenerating family of complex algebraic varieties).

The argument uses the base change compatibility and regularity of the Gauss-Manin connection to reduce the problem to the following statement: if $M$ is a complex square matrix such that, for every field automorphism $\sigma$ of $\mathbb{C}$, $\exp(2\pi i M^{\sigma})$ is conjugated to a matrix with integer coefficients, then $\exp(2\pi i M)$ is quasi-unipotent. This in turn is a simple consequence of Gelfond-Schneider theorem!

I always found this proof quite surprising and I was wondering if there aren't other unexpected applications of transcendental numbers out there.