Matrix[1] has an encryption scheme based on the Signal double-ratchet (which was based on the OTR scheme) called Olm, and is completely federated with an open standard. So you can run your modified version, and be the only person who keep a history of your conversations (only the homeservers that participate in a conversation store the chat history).

Matrix also has features like bridges to other chat systems, as well as a plethora of clients thanks to being an open standard (the most popular is Riot).

[1]: https://matrix.org/

Signal is also effectively limited to smartphones due to the phone number requirement, making it even less free.

You can use Signal desktop - all you need is a phone to receive a confirmation text message, but that needn't be a smartphone.

Even that is too much to give to a chat application. In most cases a phone number is tied to a person, requiring that is pretty much requiring a full name.

I guess one could sign up with a Google Voice number or something, but the whole idea of requiring a phone number for anything not related to a phone is just ridiculous to me.

Even more ridiculous when Walmart almost wouldn't sell me tires because I didn't have a phone number. It took three different managers to figure out how to put in 000-000-0000.