I was looking for a way to export the complete feed (all the incoming elements) from Google Reader. I rather quickly found out that you need the session cookie and that trying to use basic auth wasn’t going to cut it. I found a rather old post about the topic, but none of the advices given there seems to work.
What I discovered however, is that you can go into your Google Reader –> Settings –> Folders and Tags and make certain folders / tags public. This will generate a publicly accessible (ie without authentication) page / RSS feed. This is exactly what I was looking for!