More than two years after http://meta.math.stackexchange.com/q/4569/, this may be the time for reconsideration. A brief update:
In July 2012, when the aforementioned discussion took place, the site received $11000$ questions. In October 2014, the number was $41000$. (Source) [In addition to the above numbers, about $500$ questions asked in July 2012 have subsequently been deleted, compared to $2600$ from October 2014. (Source: special moderator search abilities.)]
In 2012, only Stack Overflow disallowed questions by unregistered users. By now, several other sites followed: Physics, Electrical Engineering, Programmers, and Ask Ubuntu.
Unregistered users sometimes lose access to their accounts after asking. So they lose ability to edit or comment on their questions, which creates noise: updates and comments posted as answers, or a new identical copy of the question posted. All this takes the time and energy of reviewers, editors, and moderators; and the users themselves are not happy either.
Split accounts are a known problem for the closure of exact duplicates.
Unregistered users cannot delete own posts. So instead they deface or blank it, leading to chaos: votes to close as unclear, rollbacks, votes to close with custom reason "OP wishes to delete", votes to delete, flags for moderator attention.
The quality... well, you know. Back in the days, Qiaochu Yuan wrote
StackOverflow has less of a need for growth and more of a need to improve the quality of content, but I don't think we're there yet.
I think we may be there now. Registration requirement will not magically improve the quality of questions, but if there are fewer of them overall, the arriving questions can be attended to more thoroughly.