  Many generalizations of continued fractions, where the reciprocal function
has been replaced by a more general function, have been studied, and it is
often asked whether such generalized expansions can have nice properties. For
instance, we might ask that algebraic numbers of a given degree have periodic
expansions, just as quadratic irrationals have periodic continued fractions; or
we might ask that familiar transcendental constants such as $e$ or $\pi$ have
periodic or terminating expansions. In this paper, we show that there exist
such generalized continued function expansions with essentially any desired
behavior.
