The way I understand this, cryptocurrencies don't need to be on blockchains at all. Whether there are any that exist off chain, I don't know.

The whole reason for crypto, though, is to make transactions decentralized, so that no one server, person, or entity, has control over them. This blockchain ledger system is more of a means to breakdown decentralization as it is anything else.

As it is, I've read that blockchains aren't nearly as fast or efficient as other solutions, so it's not necessarily the transaction speeds or the ease of scalability. It all seems to fall back on keeping things out of the hands or control of any one entity.

And that's what the blockchain is really good at—taking multiple servers scattered over the world running the blockchain, making sure each block is good, and then verifying or signing off on those blocks, thus ensuring the safety and security of the transactions on the blockchain.

It also makes it harder to hack, and thus less vulnerable to outside attacks, with authority scattered rather than all in one place.

So, a crypto could be placed on something off of a blockchain, but the question would be, why? To what purpose, if crypto exists to essentially tear down the current barriers that keep people, anywhere in the world, from quickly, safely, and cheaply, transacting with one another with little to no governing authority.