Solving or approximating infinitely nested integral Let $f$ given by
$$f(x) = g(x) + \int_0^x\left(g(x_1) + \int_0^{x_1} \left( g(x_2) + \int_0^{x_2} \ldots d_{x_n} \ldots \right) d_{x_2} \right) d_{x_1}$$
where $n \rightarrow \infty$ and $g(x)$ is strictly decreasing in $x$.
How can such an integral be solved or approximated for?

If $$h_n(x):=g(x) + \int_0^x\left(g(x_1) + \int_0^{x_1} \left( g(x_2) + \int_0^{x_2} \ldots d_{x_n} \ldots \right) d_{x_2} \right) d_{x_1} $$ converges to a limit function $h$, then the self similar nature of this expression implies that
$$g(x)+\int_0^x h(t) \mathrm{d}t=h(x). $$
Differentiating gives
$$g'(x)+h(x)=h'(x), $$
and using the integrating factor $e^{-x}$ gives
$$e^{-x} g'(x)=(e^{-x} h(x))'. $$
Thus
$$e^{-t} h(t) \big|_0^x=\int_0^x e^{-t} g'(t) \mathrm{d}t $$
or 
$$h(x)=e^x g(0)+e^x \int_0^xe^{-t} g'(t) \mathrm{d} t .$$
Finally, an integration by parts gives
$$h(x)=g(x)+\int_0^x g(t) e^{x-t} \mathrm{d}t. $$
