Acquia Dev Desktop MySQL database troubleshooting on Mac OSX 10.11
Manually starting the MySQL server:

/Applications/DevDesktop/mysql/bin/mysqld --defaults-file=/Applications/DevDesktop/mysql/my.cnf

Viewing the log:

tail -50 /Applications/DevDesktop/mysql/data/mysql.err

If everything went right you should see the MySQL green dot in the Acquia Dev Desktop GUI.