  We describe maximal, in a sense made precise, analytic continuations of germs
at infinity of unary functions definable in the o-minimal structure R_an,exp on
the Riemann surface 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 of the valuation ring of all polynomially bounded definable
germs.
