If you buy non-DRM books from an online bookstore that uses ePUB format, you can read it on your Kindle but Amazon won't have control over it. I believe both Kobo and Google Play Books use that format, and there are likely many others.
You can also buy direct from authors, and many (like myself) sometimes give away free copies.
The key is knowing how to sideload books on your Kindle. I use BookFunnel to help my readers load books on their preferred device.