Returning HTML through an ajax-call in Drupal 8
Since we are fairly new in the Drupal 8 jungle I'll give some feedback how I fixed this. Be aware this is probably not the best solution. 

Using a controller I included this function:

    /**
     * Inspiration page action
     */
    function inspiration(){
        $twig = \Drupal::service('twig');
        $template = $twig-&gt;loadTemplate(drupal_get_path('module', 'experimental') . '/templates/inspiration.html.twig');
        echo $template-&gt;render([]);
        die;
    }


This works fine, allthough it is not very beautiful.