We all own two-way hardware: computers and phones that send and receive data. These devices could relay data if there was a way of coordinating the supplies/demands of relay-resources (in particular, solving the free rider problem -- as was brilliantly good-enough-solved by Bittorrent, and somewhat solved in TOR -- ).
Bitcoin, with cheap, private, micropayment channels, is the perfect solution to this coordination problem. From there, we're just a few market-heuristics (phones automatically  estimating their ability to provide a 'useful relay' and  auctioning off these relays) away from a complete solution.
If Bitcoin could do this, it alone would enable a cheap, but more importantly, private/censorship-proof internet. Anyone who wanted to use this network would need (a small quantity of) Bitcoin.
So I have long believed that this 'global data transfer market' is the true "killer app" for Bitcoin, which would give it undeniable intrinsic value and force it to inevitably take over the planet.
Someone claims to be working on that solution: http://www.bitmesh.network/
Of course, they haven't actually submitted anything (as far as I can tell).
I certainly hope that someone works out these details soon.