Neural networks are great for working on images, so one area of web applications you could use AI for would be identifying and/or manipulating patterns in images over large sets of data. For example, a site like Flickr or Facebook might have some interesting training material to identify people based on face or associating groupings of pixels (those being the features you work with) with certain items mentioned in captions or tags.
In terms of text manipulation, there's a lot of stuff, but it's usually icing on the cake for other web apps. I'm talking mostly in the areas of automatic completion in search bars and back-end things the user doesn't usually see, like automatic machine translation or improved search capability.
The problem with putting AI at the front of an application's offering is that usually, artificial intelligence is not a feature in and of itself, but rather a way of negotiating large data sets effectively without regular prompts from the designer. In general, a user will associate with an application on a one-to-one basis, and therefore judges it only on the quality of a relatively low number of responses.