This weird (and maybe not very useful) question is inspired from generalising naively the notion of double and triple integrals to countably infinite integrals
Consider the following integral:
$$\int^{(R)}f(x)d^Rx=\underbrace{\int \cdots \int}_{\textrm{R times}} f(x)d^Rx$$
One simple example that converges is $f(x)=e^x$ with inital condition the constant sequence $(0)$, thus all integration constants will become zero
This will give us the following sequence of functions
$$\begin{matrix}(S)=e^x & e^x & e^x & e^x & \cdots\end{matrix}$$
which is "constant" in the sense that it is always the same function unchanged (since $e^x$ is an eigenfunction (modulo the arbitrary constant) of the integral operator, with eigenvalue 1). Therefore
$$\lim_{i\to\infty}S_i=e^x$$
What mathematical tools is most suitable for analysing the convergence of "functional sequences"and couuntably infinite iterated integrals in general?