It is clear that if K is a subfield of the complex field, and if f(z) is an analytic function germ which is algebraic over the field of rational functions K(z), then f may be analytically continued along any path which avoids all those complex numbers which lie in the algebraic closure of K. (We are mainly interested in the case that K is countable, so "most" paths have this property.) I discuss the corresponding situation for function germs f(z) satisfying certain transcendental equations (eg exponential polynomial equations). The methods used are most naturally expressed in the language of nonstandard analysis and the deeper results require a knowledge of o- minimal structures. However, I shall work out the details of a simple case where this background is not explicitly required. I shall then give some applications to definabilty theory for expansions of the complex field by certain analytic functions.