This is a short tutorial on enabling/disabling 'timed login' in Ubuntu Gnome 13.04
In essence, the issue is that after disabling 'automatic login' on User Accounts, the default user would still log in after 10 seconds of the login screen showing--known as 'timed login'--, which to me is a privacy issue as it allows anybody with access to the computer to log in as such default user. For some weird reason the option to disable this is not present in the User Accounts GUI even though it'd just be adding a checkbox (it wouldn't alter the design or anything)...
One possible workaround would be to create another user with no sensitive data and make that the default, but this is not ideal, at least in my case, so I figured I'd share here the simplest solution I've found in case it helps anyone out there.
All we need to do is edit as superuser the file custom.conf that is located in /etc/gdm with our text editor of choice. In my case, I used nano (a nifty terminal-based text editor):
Find the line that says 'TimedLoginEnable=true' and change it to 'TimedLoginEnable=False', save the changes and reboot the machine, you should be good to go.
sudo nano /etc/gdm/custom.conf
Hope this helps!