The master theorem seems to fail on nonlinear recursive functions. Is there a standard tool for finding the closed forms of recursive functions of this form?

The question comes from trying to find the closed form of the following recursive function: $f_i(X) = (f_{i-1}(X)^2 + f_{i-1}(X))/2$
Where:
$f_0(X) = X$

I would be willing to part with recurrence relations for this function, but I would be much more delighted to learn a general method or trick which makes finding closed forms of functions like this simple.