I'm building a 'taxi tracker' app, for want of a better term- gives you estimated trip times, allows you to share the journey with other people so they know where you are.

That part doesn't require PostGIS as such, but it's for an NYC city government app competition, so we have all sort of city datasets to use. I've used PostGIS to help make custom map tiles (preview at https://twitter.com/#!/taxonomyapp/status/149565007384940545), highlight the outline of the building you're heading to... all sorts. It's been a fantastic learning exercise.