One of the beautiful things about the Flickr API is it enables interesting things without the Flickr team having to do the work.
Flickr has no "most popular photos" page, but there's a whole "most favorites" user group. For instance, there's a recent list of the top 60 photos. The #1 photo on Flickr is a technically amazing motion shot, with 394 favourites votes since Nov 2004. It's picked up 115 favorites in five weeks. The problem with favorites lists is they're self-reinforcing. More people see the photo, mark it favorite too, and before you know it you have a boring ol' power law. "New favorites" would be more interesting.
GustavoG has done some serious social network analysis. His FOAF plot shows two major clusters that are only sparsely connected; another view shows distinct clusters that seem consistent with the history of Flickr. He's also drawn the social center of Flickr, a scatterplot of who has lots of mutual contacts, and an interesting graph that shows the imbalance of incoming vs. outgoing links. That last graph shows lots of aspirational linking to Caterina and Stewart, the Flickr founders, and points out popular people I should know about like lightpainter.
There's a wealth of data in Flickr. The API enables all sorts of volunteer analysis.