There are projects like FoldingCoin.net and CureCoin.net, which enable people to contribute processing power for solving computation-heavy problems, such as protein folding. These two projects are focussed on medical research and the tech is based on bitcoin blockchain.

To encourage people to contribute processing power / storage, these projects need to incentivise them which is where bitcoin comes in.

There is a startup called 21.co which is creating an embeddable bitcoin mining chip, which can be "embedded into an internet-connected device as a standalone chip or integrated into an existing chipset as a block of IP to generate a continuous stream of digital currency for use in a wide variety of applications." [1]

As mobile devices start having these embeddable mining chips, people would be able to "mine" coins like FoldingCoin and CureCoin on their devices, which then becomes like a network of coordinated "cloud servers" which are incentivised by these coins

Also, there are decentralised cloud storage projects like Storj.io (similar to Dropbox).

[1] https://medium.com/@21dotco/a-bitcoin-miner-in-every-device-...