Let $c(n)$ be the number of iterations of the collatz map needed to reach $1$, or $\infty$ if it never reaches $1$ (i.e. $c(n)$ is the total stopping time of $n$).
Now define $$C(n) = \max_{1 \le m \le n} c(m)$$
This will be a non-decreasing function (by definition), and if the Collatz conjecture is true, $C(n) < \infty$ for all $n \in \mathbb N_+ $.
Is anything known about the growth rate of $C$, both assuming the collatz conjecture, and in general?