Cheapest? The most expesive part is going to be paying the person or company who sets it up, maintains, and documents it. I'm an expert at Drupal, so for my companies, Drupal is cheap and it can be adapted to grow and change over time as new ideas for the application are discovered; you don't know what your website will need next year so it's nice to have a CMS that is based on having the ability to adapt over time. Wordpress is only cheap if your ambitions for the site aren't big. Trying to do too much with it is a developer's nightmare. Peace.

Oh, and unlike a lot of other answers, Drupal has great support for being headless, like you asked for.