You did not limit the context of continued fractions to numbers. Did you ? Then continued fractions can be used whenever you have a Euclidian division, preferably when there is a natural choice of quotient / remainder, so that it is done in a unique way. An important example is that of polynomials. Then continued fractions can be used to find accurate approximations of smooth functions by rational fractions about a given point, say $x=0$. This is related to Padé approximants.

This is described in the French wikipedia page (sorry, not in the English one) link text