If you're going to add (full-page) caching, Batcache [0] is the best solution. Combined with a Memcache plugin [1], you can use this to store your page caches in AWS Elasticache.
You really can't use WP without some form of caching (be it a plugin or proxy).
I do wonder how much energy is wasted from poorly optimised websites & systems?
The root post of this thread is pointing out that you should always run WP behind a CDN like CloudFlare. That will handle all of the caching that you need. If you're running a truly static site, then you're going to have very few actual hits to your server.