I created a custom field as in field_example described, using text_format as type and it shows up a multi line editor without any problems.

But I'm facing a mysql error when trying to save a node:

messagePDOException: SQLSTATE[21S01]: Insert value list does not match column list: 1136 Column count doesn't match value count at row 1: INSERT INTO {field_data_field_test3} (entity_type, entity_id, revision_id, bundle, delta, language, field_test3_additional_info) VALUES (:db_insert_placeholder_0, :db_insert_placeholder_1, :db_insert_placeholder_2, :db_insert_placeholder_3, :db_insert_placeholder_4, :db_insert_placeholder_5, :db_insert_placeholder_6_value, :db_insert_placeholder_6_format); Array ( [:db_insert_placeholder_0] => node [:db_insert_placeholder_1] => 12 [:db_insert_placeholder_2] => 12 [:db_insert_placeholder_3] => article [:db_insert_placeholder_4] => 0 [:db_insert_placeholder_5] => und [:db_insert_placeholder_6_value] => test [:db_insert_placeholder_6_format] => filtered_html ) in field_sql_storage_field_storage_write() (line 451 of /var/www/demo/site/modules/field/modules/field_sql_storage/field_sql_storage.module).

I have the following files: si_media.install:

<?php

/**
 * Implements hook_field_schema().
 */
function si_media_field_schema($field) {
  $columns = array(
    'additional_info' => array(
      'type' => 'text',
      'not null' => FALSE
    ),
  );
  $indexes = array(
    //'additional_info' => array('additional_info'),
  );
  return array(
    'columns' => $columns,
    'indexes' => $indexes,
  );
}

and si_media.module:

<?php
/**
 * Implements hook_field_info().
 */
function si_media_field_info() {
  return array(
    // We name our field as the associative name of the array.
    'si_media_additional_info' => array(
      'label' => t('Additional Informations'),
      'description' => t('Adds more informations to a media field.'),
      'default_widget' => 'si_media_text',
      'default_formatter' => 'si_media_simple_text',
    ),
  );
}

/**
 * Implements hook_field_validate().
 */
function si_media_field_validate($entity_type, $entity, $field, $instance, $langcode, $items, &$errors) {
  foreach ($items as $delta => $item) {
    if (!empty($item['additional_info'])) {
      /*if (! preg_match('@^#[0-9a-f]{6}$@', $item['additional_info'])) {
        $errors[$field['field_name']][$langcode][$delta][] = array(
          'error' => 'si_media_invalid',
          'message' => t('Color must be in the HTML format #abcdef.'),
        );
      }*/
    }
  }
}

/**
 * Implements hook_field_is_empty().
 */
function si_media_field_is_empty($item, $field) {
  return empty($item['additional_info']);
}

/**
 * Implements hook_field_formatter_info().
 */
function si_media_field_formatter_info() {
  return array(
    // This formatter just displays the hex value in the color indicated.
    'si_media_simple_text' => array(
      'label' => t('Default output'),
      'field types' => array('si_media_additional_info'),
    ),
  );
}

/**
 * Implements hook_field_formatter_view().
 */
function si_media_field_formatter_view($entity_type, $entity, $field, $instance, $langcode, $items, $display) {
  $element = array();

  switch ($display['type']) {
    // This formatter simply outputs the field as text and with a color.
    case 'si_media_simple_text':
      foreach ($items as $delta => $item) {
        $element[$delta] = array(
          // We create a render array to produce the desired markup,
          // "<p style="color: #hexcolor">The color code ... #hexcolor</p>".
          // See theme_html_tag().
          '#type' => 'html_tag',
          '#tag' => 'p',
          '#attributes' => array(
            'style' => 'color: ' . $item['additional_info'],
          ),
          '#value' => t('The color code in this field is @code', array('@code' => $item['additional_info'])),
        );
      }
      break;
  }

  return $element;
}

/**
 * Implements hook_field_widget_info().
 */
function si_media_field_widget_info() {
  return array(
    'si_media_text' => array(
      'label' => t('Text area (multiple rows)'),
      'field types' => array('si_media_additional_info'),
    ),
  );
}

/**
 * Implements hook_field_widget_form().
 */
function si_media_field_widget_form(&$form, &$form_state, $field, $instance, $langcode, $items, $delta, $element) {
  $value = isset($items[$delta]['additional_info']) ? $items[$delta]['additional_info'] : '';

  $widget = $element;
  $widget['#delta'] = $delta;

  switch ($instance['widget']['type']) {
    case 'si_media_text':
      $widget += array(
        '#type' => 'text_format',
        '#default_value' => $value,
      );
      break;
  }

  $element['additional_info'] = $widget;
  return $element;
}

/**
 * Implements hook_field_widget_error().
 */
function si_media_field_widget_error($element, $error, $form, &$form_state) {
  switch ($error['error']) {
    case 'si_media_invalid':
      form_error($element, $error['message']);
      break;
  }
}

I don't get it, why there is this error coming.