Run WordPress without a server is an option today
The concept of a server is about outdated. Why do you need a server?
Running the code on Google App Engine, Amazon Lambda, Microsoft Azure Functions, IBM Bluemix OpenWhisk or another stateless solution seems much more sensible and logical.
Google App Engine
Currently, you can run WordPress on Google App Engine. When you Google the term there are many how to´s and useful information. In the WordPress plugins directory, several solutions will ease the transition to serverless computing. Several companies are running production WordPress sites for customers on Google App Engine. There is still some limitations so it might be a bit early to make the transition.
Amazon Lambda
Guess it’s just a matter of time before WordPress runs on Amazon Lambda as a serverless and stateless app.
Microsoft Azure Functions
Another alternative is to run WordPress on Microsoft Azure Functions. The current WordPress runs better on Unix-like operating systems, but this might change.
IBM Bluemix OpenWhisk
The serverless alternative from the computing giant IBM is another option.