We describe maximal, in a sense made precise, L \mathbb {L} -analytic continuations of germs at + ∞ +\infty of unary functions definable in the o-minimal structure R an,exp \mathbb {R}_\textrm {an,exp} on the Riemann surface L \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 R poly \mathcal {R}_{\text {poly}} of the valuation ring of all polynomially bounded definable germs.