Here are a few examples of blockchain technology being used to blaze new trails in areas other than currency transactions:
Namecoin was the first of the altcoins, but it could also be considered the genesis of the "blockchain beyond currency" movement. Namecoin's primary function is for regstering domain names for the .bit domain, which are recorded on its blockchain- creating a decentralized DNS. Namecoin itself is used to pay the fee for these registrations.
Namecoin is based on the Bitcoin code base, and it was the interest in using the blockchain to store data other than currency transactions that led to it becoming a separate entity with its own blockchain, rather than an update to Bitcoin itself.
Ethereum is a development platform that can be used to build decentralized applications or currencies, and applications based on "smart contracts". Ethereum uses its underlying network unit, Ether, as payment to execute Ethereum contracts.
Storj is a decentralized cloud storage platform that uses blockchain technology to keep track of files. In much the same way that people "mine" bitcoins by volunteering processing power, Storj users earn "SJXC"- the coin used by Storj- by setting aside space on their hard drive to be used for cloud storage on the Storj network.
MaidSafe is a decentralized storage platform, similar to Storj- but it goes beyond just offering cloud storage. MaidSafe is more like a server environment than a cloud storage system. It is intended to be used as a platform for building decentralized applications. A few examples given are web sites, secure messaging, virtual wallets, and document processing. The word "MaidSafe" is actually an acronym, which stands for "Massive Array of Internet Disks - Secure Access For Everyone".
Alexandria is an open-source, decentralized library & market for media of any kind, protected by a blockchain. Like YouTube or Spotify with no servers, ads or censorship.
Factom maintains a permanent, time-stamped record of your data in the blockchain, helping to reduce the cost and complexity of conducting audits, managing records, and complying with government regulations.