Personally, I have like 40-45 pages or Inbox/Outbox PM's. If we do the math it's like 1700-1800 messages combined (20 PM per page). And I have seen others saying, they have much much much higher amounts of PM's saved up.
Quoting for visibility, just look below. Man that's a lotttttt of PM we are talking about. I want to believe it has no limits. And even if it did, it's set so high that we can't reach it normally, unless we use some tools to spam the PM inbox/outbox.
I have ~1,400 pages of PMs in my inbox. I’ve learned to cope with the PM times. I have no desire to store them on my own.
I used to always delete PMs as I received them or I’m sure I’d have over 2,000 pages. Somewhere around 2016 I lost control.
Btw, after poking around a little, I found this one.
This answers your question (Hopefully). 
I've looked into setting that option by default, but it wasn't obvious to me how to do it. It looks like it might be difficult.
There is no PM limit.