As others note, SQLite is a consideration, although you will want to be aware of limitations:
http://www.sqlite.org/limits.html
If you are on a Windows Server system (not likely, but possible), you can use Access, and connect with ODBC:
http://www.w3schools.com/PHP/php_db_odbc.asp
Also, maybe Firebird:
http://www.firebirdsql.org/
However, just note some things before you get too far into it.
- File locking - having multiple users at once can lock out request responses until the locks are cleared
- File access - if you put a file within your HTML directory, you will need to take precautions against allowing other people to download you database; best to keep it off your public path altogether
- Optimization - MySQL and PostgreSQL are going to work much better; none of these other options are going to scale gracefully, which leads me to my last note...
- Undoing what you've wrought - When you get your sites large enough to require a better database system, you could find yourself needing to redo a bunch of code or finding a way to port your data to a new setup
Seriously consider whether you can store it all together, or maybe look into another provider which can help make it more affordable. For instance, I've liked Dreamhost ($8.95/month for 5 databases), although my favorite provider is MediaTemple, who I used for several years and cost about $22/month. There are others out there that are similarly competitive, even Amazon Web Services.