I see a lot of people saying they are getting auth prompts for public anonymous content on SharePoint sites. Once you have anonymous enabled in Central Admin and in the site collection, you'd expect it to be all good right?
I discovered one more resaon beyond the obvious permissions problems. If you put a graphic or something embded in a page, then publish that page but NOT the embeded object (i.e. it's still draft or unapproved) you'll get an auth prompt when the page loads.
On a busy (lots of stuff) page it can be tough to find what's the issue. The quickest way I've found to discover the problem (99% of time an image) is to cancel out of auth prompts, then look for broken stuff on the page. An image you can right click the broken icon and find the location of it... then jump to that library and check the file. My bet is it's never been published.