Recall continued fractions: http://en.wikipedia.org/wiki/Continued_fraction

Now take a look at this question: https://math.stackexchange.com/questions/601846/the-limit-of-displaystyle-lim-n-to-infty-exp-1-exp-2-exp-3-ldots-exp/601890#601890

You cannot help but notice that there is some resemblance between recursively applying something like 1/x:

$ x \to a_n + 1 / x $

and recursively applying something like exp(-x):

$ x \to a_n + \exp( -x ) $

It may be worth investigating the propertied of the map

$ x \to a_n + f( x ) $

for functions $f$ that are monotone asymptotically decreasing to $0$ to ensure convergence.

Thus the question: were such things investigated before, apart from generalized continued fractions?