We use generalized power series to construct algebraically a nonstandard model of the theory of the real field with exponentiation. This model enables us to show the undefinability of the zeta function and certain non-elementary and improper integrals. We also use this model to answer a question of Hardy by showing that the compositional inverse to the function (log x) (log log x) is not asymptotic as x→+∞ to a composition of semialgebraic functions, log and exp.