When you create a new droplet your password is emailed to the email address on your account unless you have opted to use ssh keys. If you select one or more ssh keys on the create screen no email is sent and no default password is sent since it is not required to access the server using your keys.
Sorry, I just realized that I wasn't being very clear when I wrote my question... LAMP has its own password (part of MySQL), and I cannot log into it. Typically when installing LAMP on your own a dialogue requesting a password comes (but when using the Digital Ocean auto-install tool, this does not occur).
Yes. The password you would set up when installing mysql-server on ubuntu is the root MySQL password. This is displayed in the MOTD on your droplet when you log in via ssh. An example is below:
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Thank you for using DigitalOcean's LAMP Application.
Your web root is located at /var/www/html and can be seen from http://104.131.61.241/
The details of your PHP installation can be seen at http://104.131.61.241/info.php
Your MySQL root user's password is vxQO7NRuEl
You are encouraged to run mysql_secure_installation to ready your server for production.
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To delete this message of the day: rm -rf /etc/motd.tail