Hmmm…. Not very nice this one.
OK – so we switch to using my-huge.cnf or my-large.cnf. On current MySQL – Debian Jessie – this changes a couple of settings.
One – it turns on binary logging by setting:

log-bin=mysql-bin

in the cnf file.
BUT – IT DOES NOT SET UP LOG ROTATION!
So, the log files will get created – and more and more will be created – until no space is left.
Fairly easy to fix.
In the my.cnf file add a:

expire_logs_days        = 1

then restart MySQL. This should set the log files to be deleted safely after one day – and the restart will carry out a purge of the current binary log files. (I also like to manually restart to check it restarts OK – i.e. to check that I’ve edited my.cnf correctly).
Also – on my system it cleaned out approx 140GB in a couple of seconds. Obviously, the following commands were carried out after editing my.cnf.

root@server:~# cd /var/lib/mysql/
root@server:/var/lib/mysql# du -sh
153G .
root@server:/var/lib/mysql# /etc/init.d/mysql restart
Restarting mysql (via systemctl): mysql.service.
root@server:/var/lib/mysql# du -sh
9.6G .