Yes, I have seen doctors and dentists ask for the SSN, and categorically refuse to provide it.

Unless you personally are nearly all Americans, good for you, but not relevant to OPs post about "Very Secret Number"

And none of them have ever complained when I left if blank.

I've seen forms that explicitly say to put in all nines if you "don't have one", so that's what I do everywhere that insists on asking but doesn't have a legitimate purpose (ie tax reporting). To any human it should be obvious that all nines indicates an exception.