If you want to get fancy you can even attach it to your domain root (example.com vs www.example.com) using Route53. Which is impossible with many static hosts. Although that requires a hosted Route53 zone which at $2 might very well be 100x your hosting costs.
I use Jenkins to generate the website itself.
Edit: Only downside is if your traffic spikes you have no control over the cost. There is no upper bounds. With that said, it would take a tremendous amount of traffic to balloon the costs to anything worth worrying about. And at least you can be sure your website will actually stay up.