Demo of launching a binderhub notebook server with a free running Postgres server.
Postgres is installed via Anaconda; the databases is initialised into /srv/conda/pgsql (the /src/conda (and if it exists /srv/npm) path is the only writeable path outside of $HOME).
The init_db/postgres/init.db file is called to set up any required users, databases, etc, and it in turn seeds a test database as defined in init_db/postgres/seed_db.sql. The init_db directory is also copied into /srv/conda so the user can check on the initialisation set-up if required.
Example via @manics on Jupyter dscourse site [gist].
See the other branches for versions that do, albeit at the expense of using a Dockerfile.
See also: https://blog.ouseful.info/2019/02/04/running-a-postgresql-server-in-a-mybinder-container/