For example, a distributed blockchain that would keep track of the weather. Anyone could add a weather reading, and several of these would be stored in each block, just like transactions in a cryptocurrency. There is no reward for mining these blocks, so speed of mining wouldn't be a huge concern for inflation, all that would matter is that all the nodes agree.
It seems to me that Proof of Work would work for this, but it would be lots of effort for nodes to get nothing in return, also the benefit of slowing mining isn't really important. Proof of Stake seems to be the next most popular consensus mechanism, but I can't see how that could possible work in this situation, where you have nothing of value to risk.
What other methods of consensus could work for this?