Is there a list of all the fonts which come with OpenOffice / LibreOffice ?
Partly answering my own question:
OpenOffice includes OpenSymbol, DejaVu,the Liberation fonts (from 2.4 to 3.3) and the Gentium fonts (since 3.2). Versions prior to 2.3 included the Bitstream Vera fonts. After 3.4 the GPL-licensed Liberation fonts were removed and replaced by the Apache-licensed ChromeOS fonts Arimo (sans serif), Tinos (serif) and Cousine (monospace). OpenOffice will also use the default fonts of the running operating system.
Fonts are inherited by from the OS, not the application. If you want to know what fonts you will have in any product, open your fonts folder.
Programs adding fonts is not common, especially for something where the file needs to be shared and still look good.
Photoshop is a good exception, but OpenOffice plays it safe:
How can I add fonts to OpenOffice?
OpenOffice.org uses the fonts that are installed in your system. Add new fonts using the Font management tools for your Operating System (for example in Microsoft Windows, you add new fonts via Control Panel
Fonts). You may need to restart OpenOffice.org (including the Quickstarter), to see the new fonts in the Font Name list box.
And, LibreOffice also seems to play by the rules:
The set of fonts is not property of the office suite, but only the configuration of your operating system. Install the necessary fonts in the system and they will be available in office applications.