Q:

Is this property of the Collatz sequence interesting? 

As an amateur playing around with the Collatz conjecture, I've stumbled on something I haven't seen mentioned before, and that may or may not be noteworthy. 
Suggested by Gottfried Helms, here's a more intelligible version of my discovery:

Let $s(n)$ denote the total stopping time of $n$ under the original Collatz map, and consider the iteration $n_{k+1}=s(n_k)$ . 
  I noticed that the sequence of $n_0,n_1,n_2,...$ seems to converge to $1$ for all $n_0>1$ - that is, repeated application of $s$ will always yield $1$ eventually. 

At least this was the case for every $n$ I tried.  Examples include $42 \to 8 \to 3 \to 7 \to 16 \to 4 \to 2 \to 1$ and $9 \to 19 \to 20 \to 7 \to 16 \to 4 \to 2 \to 1$.
What I'd like to know is this:

Can it be proven that this property holds, or doesn't hold, for all $n>1$? 
Is the same true for other mappings in the Collatz "family"?  More generally and vaguely, is this anything interesting at all? 

A:

In terms of your function $s$, the Collatz conjecture can be rephrased:

Conjecture: $s$ is defined on all of $\Bbb N$.

Without having proven the Collatz conjecture, you do not even know if your map is defined on all of $\Bbb N$. If your map is defined, the Collatz conjecture is true.  If the Collatz conjecture is true, your map is defined.  So the existence of this function is equivalent to the Collatz conjecture. 
In order to study conclusions about special properties of the map, you'd have to assume the Conjecture is true. 

A:

$s(5) = 5$, and there seem to be lots of starting points that end up at $5$, but I couldn't find any other cycles up to $1000000$.  The sequence seems to decrease much faster than Collatz itself (the largest number I could find with $s(x) > \sqrt{x}$ is $x=115547$) so I doubt any more cycles or run-away sequences can be found by searching.