In October 2016, Grace noted that registration to ask a question had been implemented on math.stackexchange. I would like to know whether this has had any effect on our site.
The registration requirement had a big impact on the issues related to users losing access to their unregistered account (points 3,4,5 in the feature request).
It is now a rare exception that a user needs to be told about merging their accounts. Before the change it was a common occurrence that a user posted a question from an unregistered account, later created a new account (often but not always), but lost their cookies so that they didn't have access to their first account any more, and then posted clarifications to their questions in response to comments - or requests for elaboration on a point in an answer - as an answer from the new account.
It still happens that users post non-answers instead of editing their question or posting a comment, but that's much rarer than it used to be.
Also deletion attempts by defacing or blanking the questions are now rarer, though their number didn't decrease as much as the other issues. That's not surprising, since such attempts are often made when the asker can't delete the question themselves anyway due to answers posted. Also, many users use the android app which doesn't provide a delete link, and thus don't have an obvious way to delete their post even when they technically can.
I don't see any big effect on anything else, except of course a big increase of the number of newly registered accounts [per time unit] since then.
It is of course possible that the growth would have been larger without the change, but the site growth had already slowed before.