If you can't log on from the launcher, try double-clicking EverQuest2.exe in your EverQuest II folder (you can go there by right-clicking EverQuest II launcher icon and choose "Open file location"). I could log on that way.
Can confirm. "There was an error completing your request. Please try again." when logging in through the launcher.
open up the EverQuest II directory, launch directly everquest2.exe, input username/passphrase,.. instant connection.
Maybe this is longer than an hour (tweet from 5 hrs ago) Tweet on the forum homepage says "undergoing maintenance that would be at least another hour to complete" due to the issues of logging in.
Works indeed. If you can't remember how to spell the character's name you wish to login, check the same folder you found the .exe in for an .ini named <account>_characters.ini and it'll list all your toons within.
Make sure you use Everquest2.exe and not EQ2.exe which will only bring the usual login dialog up and fail. Just worked again for me even though the launcher is still failing.