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Do you suppose there might be a finite number of highest numbers?

No, but I have found after running several million tests, that there are major clusters similar to 9232 found in higher regions. One thing that I think is cool is that with the exception of numbers that are powers of 2, (powers of 2 are always the shortest route to 1 for obvious reasons) there is no easy way to guess how many steps a given number will take to get to 1 without running it.

From the list I posted previously...

25 takes 23 steps, and 88 is the highest number reached.
26 takes 10 steps, and 40 is the highest number reached.
27 takes 111 steps, and 9232 is the highest number reached.
28 takes 18 steps, and 52 is the highest number reached.
29 takes 18 steps, and 88 is the highest number reached.

The number 27 is definitely an outlier in the number of required steps.

There are other variations on the Collatz Conjecture. One multiplies by 5 and divides by 2, and has the same result. Strangely that 9232 number pops up there and in other variations as well. I did some searching on the 9232 number and couldn't find anything specific about it.

I wish I'd known about stuff like this when I was young and stupid.

Kinda bored here, so I thought I'd follow up on the 9232 number. To quantify how often that particular number shows up, lets look at the numbers between 2 and 5000. What percentage of those numbers have 9232 as a part of their 'path' on the Collatz number tree? To find out, I used a bash script that I wrote that will take as input the number to start and stop on. Unfortunately, for my purpose, it also prints out a line that says what the highest number reached was, so we have to suppress it. That's what the "grep -v reached" is for.

$ collatz2 2 5000 | grep 9232 | grep -v reached | wc -l
1996

So out of 5000 numbers, 1996, or 39.9% of them will pass through the number 9232.

Yup. That's a lot. But is it unusual? Well let's take a look at the numbers that surround it...

$ for x in 1 2 3 4 5 6 7 8 9; do
> collatz2 2 5000 | grep 923${x} | grep -v reached | wc -l
> done
0
1996
8
23
0
0
0
2
0

So, looking at the above, that translates to the following

9231 = 0
9232 = 1996
9233 =8
9234 =23
9235 =0
9236 =0
9237 =0
9238 =2
9239 =0

I'd call that unusual. What does it mean, if anything? I have no idea.