"Rocket elements" in bijections $f:\mathbb{N}\to \mathbb{N}$

Choose any infinite set $A\subseteq\mathbb N$ such that $\mu^+(A)=0$. Enumerate both $A$ and $B=\mathbb N\setminus A$ as \begin{align*} A&=\{a_1<a_2< \dots < a_n < \dots\}\\ B&=\{b_1<b_2< \dots < b_n < \dots\} \end{align*} Then you can definite the function $f\colon\mathbb N\to\mathbb N$ \begin{align*} f(b_k)&=b_{k+1}\\ f(a_1)&=b_1\\ f(a_{k+1})&=a_k \end{align*} for $k=1,2,\dots$. It is clear that $f$ is a bijection.

For this function $f$, the set of the rocket elements is either $B$ or $B\cup\{a_1\}$ and the asymptotic density of this set is equal to one.

You could also make the condition for $r$ to be a rocket element be much stricter such as $$f^{(n+1)}(r) \gt f^n(r)^{f^n(r)!}$$ for all $n.$

Pick $A,B$ as before. Define $f$ on $A$ to satisfy this strong rocket condition. You will have a countable number of wildly increasing sequences. Split $B$ into a countable number of countable subsequences in a more or less explicit manner and use them to provide infinite front ends for each increasing sequence. These front ends might be strictly decreasing, but need not be.