The functions considered are positive real for positive real $x$, and monotonically increasing to infinity. Given such a function $f(x)$, define the (nonlinear) operator $f\mapsto f^*$ by $$ f^*(x) := 1 + \frac {x}{f(1)} + \frac{x^2}{ f(1)f(2)} + \frac{x^3}{ f(1)f(2)f(3)} +\dots = 1 + \sum\limits_{n=1}^{\infty} \frac{x^n}{\prod\limits_{k=1}^n f(k)}. $$ The new function $f^*$ is again positive real for positive real $x$ and monotonically increasing to infinity. (In fact $f^*$ is an entire function, not that we need that.) So this process can be iterated. It seems that, no matter what function we start with, the result converges pointwise.