Is it possible to download paper from academia.edu without being registered there?
Google Scholar provides the direct link http://www.academia.edu/download/37277308/Introduction_to_the_Keisler_Order.pdf. It seems like there are two identifying elements: the code 37277308 and the filename Introduction_to_the_Keisler_Order. Both appear in the source of the article page on academia.edu even when you are not logged in, for instance here:

  registrationParams: {
    doc_id: 37277308,
    splash: true,
    redirect: "https://www.academia.edu/11902653/Introduction_to_the_Keisler_Order?auto=download",
  }


So in theory it should be possible to recover the direct link by piecing them together. I don't know if there is software already available, but it does definitely seem possible to write, for instance, a Greasemonkey script that adds direct download links to these pages.

Of course, all of this depends on the specific format of academia.edu's webpages and direct download links, which are subject to change at any time. In particular, if they find out that many people are using them, I suppose they will quickly change the format to pull the plug and force people to login.