1. Write down all of its proper divisors (including 1 and the number itself)
2. Add the digits of these divisors.
3. Use this sum to repeat steps 1 and 2 until your number does not change.

Must the process terminate?
At what number(s) can the process terminate at?
What numbers <1000 take the most steps to terminate?

I don't have any proof of anything, but the numerical bit is easy

I don't know if it must, but all the ones I've tried terminate.

They terminate at 15.

924 takes the most steps (21) though quite a few take 20 steps.

924 168 102 36 46 18 30 27 22 9 13 5 6 12 19 11 3 4 7 8 15