Recently a client of ours wanted to keep their site hosted on their existing Godaddy shared hosting account. There are several reasons I dislike shared hosting in general and Godaddy particularly, but I was pretty impressed that I was able to get capistrano working with WordPress on their shared hosting server.
On our own servers, we deploy WordPress via dokku. It is a heroku-style docker container manager. We can just do a git-push to the dokku server to release a new version. It spins up a new container, checks out our repo, and runs all of our build scripts. I will do a blog post soon about this, but there are a few more details I want to improve about it before I publish.