That’s where hashcash came from I believe and same principles here could be used here (using Bitcoin for relays rather than email) that it becomes uneconomical to post excessive spam. FYI @Adam Back
That works for encrypted DM type notes. Not sure about public stuff though if the relay remained in the dark about what it’s transmitting unless the message is sharded and that sounds complicated.
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Well you're not using "Bitcoin" per se, it's just adding proof of work to messages. Coracle does this. It just uses your computer to do proof of work on a hash of your message.
No reason why it couldn't work for public notes too. Nothing is encrypted.
It's the same kind of thing that Anubis does: Force the user instead of some "captcha" to do a small amount of computational work, that's no problem if it's a human but becomes uneconomical if it's a massive web scraper" 

Anubis: Web AI Firewall Utility | Anubis
Weigh the soul of incoming HTTP requests to protect your website!