Thanks, that clarifies it quite a bit. If I understand correctly; apart from my guess that the blockchain might be kept in parts by different users (turns out the whole block chain must be stored by each user), the only thing I had fundamentally misunderstood is that the calculation cost goes up for every transaction. It does not, since you only use the latest hash to calculate the next one (plus all the new data, obviously). Pretty funny, since that was pretty much the only one that mattered for the point I wanted to make (blockchains being power inefficient).
In practice, most users don't run bitcoin software at all, but interact with service providers like Coinbase. You can also run client software that only downloads part of the blockchain and allows you to generate new wallet addresses, monitor the network, and initiate transactions. Many clients are suitable for portable devices. You can even run client software on a computer that never connects to the internet, using it to create new addresses with minimal risk that the private keys will be stolen. Bitcoins don't really change hands, they are always in the blockchain, assigned to addresses. Individuals control private keys, which give them the ability to initiate transfers from an address. The power consumption is considerable and a fair source of criticism, as the low hashes produced by all that effort have no use beyond proving that you did a lot of work.