In the past few weeks I had two clients with a similar issue - both had some users expiriencing multiple requests for user name and password whenever they accessed the site. With one client entering the user name and password sometimes worked, and sometimes would request 10 times and then show the site, and sometimes would just result in a blank, empty page. With the other client, they would always get to a "internet explorer cannot display the web page" error page.
If you research the issue on your favorite search engine, you will find a lot of references to authentication providers (forms, claims and so on), registry issues, trusted sites or intranet zone settings or maybe the users are on the server and the loopback check was not disabled. Well, none of these were valid in the case of these two clients - for one thing, the users were not on the server - they were on their desktops. For another thing - even when the site was added to trusted sites it still happened. Most perpelxingly - with one client it only happenned to users on windows XP, not the users who were on Windows7...
After wasting a lot of time troublshooting the issue, I finally realised what must be different between those users and the rest, and why it sometimes worked and sometimes it didn't - it was the proxy server!
Turns out both clients had a proxy server, with proxy settings deployed to desktops via group policy. The proxy would behave differently every time it was asked for the new web site address, and would cause an authentication prompt for each image on the page. In one organisation, the proxy settings that were deployed were different for Windows XP users - which explains why only those users expirienced the issues.
The solution? I told the network admins to either add the web site to the proxy exception list in IE (via group policy) or fix the proxy server itself.