What a weird function! Here are a few scattered observations and curiosities that came to mind as I was playing around with it:

If you plot the first few thousand terms, \$$f^n(0)\$$ appears to be \$$O(\log(n))\$$:

This is consilient with the arrangement of the terms John (#2) posted: in general, it takes 2^n terms to go from 1/n to n/1. The linearly increasing subsequence \$$\frac{1}{1}, \frac{2}{1}, \frac{3}{1}\ldots\$$ occurs at times that are coming exponentially further apart, so you'd expect growth to be logarithmic in general.

For convenience, let's define \$$g(n) = f^n(0)\$$. So \$$f\$$ is a function from \$$\mathbb{Q}\$$ to \$$\mathbb{Q}\$$, and \$$g: \mathbb{N} \rightarrow \mathbb{Q}\$$ is a sequence of iterates starting at 0.

One natural question to ask is: "is \$$g\$$ surjective onto \$$\mathbb{Q}\$$?" I.e., for every \$$\frac{p}{q} \in \mathbb{Q}\$$, is there an \$$n \in \mathbb{N}\$$ such that \$$g(n) = \frac{p}{q}\$$? From numerical evidence, I conjecture "no". For example, here is a plot of the first ten thousand iterates, with the numerator along the x axis and the denominator along the y:

So not only do the iterates not appear to be filling in \$$\mathbb{N}^2\$$ in any systematic way, but the plot appears to show some kind of fractal bifurcation structure. What's up with that? Considering each rational number \\frac{p}{q}\\) as a point \$$(p, q) \in \mathbb{N}^2\$$, what is the image of \$$g\$$? (NB we haven't shown yet that \$$g(n) \geq 0\$$ for all \$$n\$$, so it's slightly unsporting to even talk about \$$\mathbb{N}^2\$$ at this point, but one gets the idea.)