How to store additional_settings value when adding a node?
Turned out I simply had to look into nodeapi_example.install to find out about hook_schema() to create a new database table where I then can save my values to. So I created an install file and put the following code inside:

function MY_MODULE_schema() {
  $schema['MY_MODULE_string'] = array(
    'description' =&gt; 'Stores information of extended content.',
    'fields' =&gt; array(
      'nid'    =&gt; array(
        'description' =&gt; 'Node ID that the string is applied to.',
        'type' =&gt; 'int',
        'unsigned' =&gt; TRUE,
        'not null' =&gt; TRUE,
        'default' =&gt; 0,
      ),
      'vid' =&gt; array(
        'description' =&gt; 'Revision ID, as we are tracking strings with node revisions',
        'type' =&gt; 'int',
        'unsigned' =&gt; TRUE,
        'not null' =&gt; TRUE,
        'default' =&gt; 0,
      ),
      'url' =&gt; array(
        'description' =&gt; 'MY MODULE string',
        'type' =&gt; 'varchar',
        'length' =&gt; 19,
        'not null' =&gt; TRUE,
        'default' =&gt; '',
      ),
    ),
    'primary key' =&gt; array('vid'),
    'indexes' =&gt; array(
      'nid'   =&gt; array('nid'),
    ),
  );
  return $schema;
}