How would you replicate the internet with only local connections and limited carrying capacity like phones? I expect something could be worked up where each user notes which sites he or she is interested in, and tries to gather and host as much of that site as possible in a local cache. Transfers are done automatically when one phone meets another, but because of limited bandwidth, limited storage space and limited time per connection, only bits and pieces would be transferred. The post-server internet would look a lot like peer-to-peer Twitter, probably with hashtag subscriptions.
The critical piece of this puzzle is opportunistic data transfer on chance meetings between strangers, such as public transport. An interesting side-effect is that most of the messages on this peer-to-peer system would be relatively local by necessity. Since most people aren't travelling between cities very often, inter-city bandwidth would be very low, and would only carry high-priority items of interest to frequent travellers. Private messaging over very long distances, in this theoretical scenario, wouldn't really work.
Mokalus of Borg
PS - I just wonder about stuff like this sometimes.
PPS - As far as I know, nobody is trying to actually do this. It would be a bit pointless.