Check the date and time in the clock on your computer: (double) click the clock icon on the Windows Taskbar.

• http://kb.mozillazine.org/Cookies

Date and time are all correct. I couldn't see anything in that cookies list which suggests any changes I can make to FF to fix the problem. I have no websites listed in the 'exceptions' list for cookies.

Start Firefox in Diagnose Firefox issues using Troubleshoot Mode to check if one of the extensions or if hardware acceleration is causing the problem (switch to the DEFAULT theme: Firefox/Tools > Add-ons > Appearance/Themes).

• Don't make any changes on the Safe mode start window.
• https://support.mozilla.org/kb/Safe+Mode

Does it help if you switch temporarily as a test to Private Browsing mode?

• https://support.mozilla.org/kb/Private+Browsing

This is getting properly weird. It seemed to work fine in safe mode, so I assumed it was one of the plugins.

I disabled all the plugins I don't use regularly (and a few which I do), but no luck. However it seems to remember me on some sites - it logs me out of youtube, and every forum, but some other websites (other than forums) which require a login it remembers absolutely fine. I am now seriously confused.

You need to disable extensions if it works in Safe-mode.

See:

• https://support.mozilla.org/kb/Troubleshooting+extensions+and+themes