Dropbox and Spotify come to mind as high-profile startups whose product depends entirely on the capabilities afforded by being on the desktop vs. just doing a web app.
Dropbox is mixed web and desktop, as a large part of its functionality is exposed (only) through the web interface. Its desktop parts are probably born more out of necessity than desire: a web app simply cannot "sync a folder".
I have never once used Dropbox's web interface (I haven't even bothered to look at it). What part of the functionality is missing from the desktop application? The only functionality I need or use is dropping files into the Dropbox folder and knowing they're not going to be destroyed if a hard drive fails or the house burns down.
