Recommended and mentioned products
Spend your time on writing high quality documentation, not on the tools to make your documentation work.
Read the Docs is the standard de-facto for serving technical documentation, especially popular among Open Source projects. It supports Sphinx and MkDocs out of the box, supports multiple versions of the documentation and localized versions. The project readthedocs.com provides commercial support and serves both public and private documentation.
Hugo is a general-purpose website framework for generating static web pages.
Create beautiful, intelligent, responsive API documentation.
Slate and friends. The next step towards a more controlled evolution of the API documentation is a tool like Slate and the API documentation in git. It's still text that you need to write manually in Markdown, but Slate provides a structure, a slick renderer, and lets you publish your API as an independent website.