Hi, one of the devs here! If you want to see a cool example of Unpaywall in action, check out the Nature paper [1] published last month about TRAPPIST-1, the nearby solar system with seven earth-sized planets. Paywalled on Nature, but Unpaywall links you to a free copy on ArXiv. [2]
[1] http://www.nature.com/nature/journal/v542/n7642/full/nature2...