Alex Y

Would it ever produce a 4 or 5?

Great question, Henry! I assumed so, and carried it six more places, but did not get a 4 or 5, I wondered if a computer whiz here could program it to go out further. But driving back from some errands today, the answer came to me:
You will always get numbers made up of 1's, 2's, and 3's only. Assume there was a larger digit in a number in this sequence, and consider the first such number. Then the previous number had at least 4 1's, 4 2's, or 4 3's. 1111 means that the number before that had one 1 followed by one 1, but that would be represented by 21. Similarly for 2's and 3's.