I work on many sites and some of my databases are too large to upload with phpMyAdmin. One of the biggest issues I have had was trying to import a large database into MAMP or take a large database out of MAMP and put it live. Another problem is when I am changing computers or restoring my Mac it is much easier to use Time Machine and grab these files instead of having to create a backup and re-import them. After failing on many searches on Google I finally found out where MAMP stores the DB files.

The path you can find these files is:
/Library/Application Support/living-e/MAMP PRO/db/mysql
For the latest version of MAMP Pro (currently 1.8.1):
/Library/Application Support/appsolute/living-e/MAMP PRO/db/mysql

I assume that if you are running MAMP Pro, you would only need to amend the path to MAMP if not.