Analytic continuations of log \log - exp \exp -analytic germs

We describe maximal, in a sense made precise, $\mathbb{L}$ -analytic continuations of germs at $+\mathrm{\infty }$ of unary functions definable in the o-minimal structure ${\mathbb{R}}_{\text{an,exp}}$ on the Riemann surface $\mathbb{L}$ of the logarithm. As one application, we give an upper bound on the logarithmic-exponential complexity of the compositional inverse of an infinitely increasing such germ, in terms of its own logarithmic-exponential complexity and its level. As a second application, we strengthen Wilkie’s theorem on definable complex analytic continuations of germs belonging to the residue field ${\mathcal{R}}_{\text{poly}}$ of the valuation ring of all polynomially bounded definable germs.