What is the asymptotic time complexity of the number of steps of "Half Or Triple Plus One" ( HOTPO)?
By request, two facts that are known and seem somewhat related to your question.


As a lower bound: infinitely many integers $n$ take time $\Omega(\log n)$. Applegate and Lagarias.
As a sort of an upper bound: $\Omega(N^c)$ positive integers bounded by $N$ reach $1$ in finite time, where $c \approx 0.84$. Krasikov and Lagarias.


Hopefully you have read Lagarias's survey before asking the question.