Public API Lists
github.com
github.com
More:
https://www.programmableweb.com/apis/directory
For example, YNAB's API is listed here, which allows a tool using the API to interact with one user's personal account at YNAB.
I'm not sure what part of that I would consider public apart from the documentation. Isn't that just "an API"?
This is just the same distinction as private versus public function signatures in a library or a system service extended to remote services. The runtime may or may not use technical means to prevent you from using the private API, but whether or not you can do it, if you choose to, beware that it is to be considered an implementation detail and subject to change at any time without prior notice.