I've discovered that a character or set of characters within password of my phpBB database user lead to access denied SQL failure, after phpBB displays the initial phpBB page successfully.
Then, I've manually emptied cache/ folder, and was able to display one forum page successfully, before getting the SQL access denied message.
Once, I've changed password – the problem goes away.
I was able to reproduce the problem by reverting to the original password.
Please note, that the original password did not cause any problem for phpBB 3.0.12.
Database: MySQL using mysqli interface.
The problematic password is "Lh3oy9Qi%%^4" (excluding double quotes).
P.S. I've reported the issue on the support forum first: