We did, when working on the JI QuickBooks module, see the code below to see how we did it.

Background information

How can we program Drupal to create a set of three (or more) fields by just clicking a button? More importantly, when this form is submitted, how is the data accessed using $form_state['values']? Easy, right? Don't go looking in the Example module. Better solution came from a stackoverflow.com post, even though it needed a little bit of work. Let's see the modified version below:

function hook_form_alter($form, &$form_state) {
  $form['agencies'] = array(
    '#type' => 'fieldset',
    '#title' => 'Tax Agency',
    '#prefix' => '<div id="replace-this">',
    '#suffix' => '</div>',
    '#tree' => TRUE,
  );

  // Build the initial fields if this is not being rebuilt 
  // from ajax request.
  if (!array_key_exists('clicked_button', $form_state)) {
    create_form($form, 0);
  }
  else {
    // Otherwise add the fields for each existing value.
    foreach ($form_state['values']['agencies'] as $i => $value) {
      create_form($form, $i);
    }

    // Add the additional fields for a new entry.
    $last = count($form_state['values']['agencies']);
    create_form($form, $last);
  }

  $form['add_more'] = array(
    '#type' => 'button',
    '#value' => t('Add More'),
    '#ajax' => array(
      'callback' => 'ajax_simplest_callback',
      'wrapper' => 'replace-this',
      'method' => 'replace',
    ),
  );

  $form['submit'] = array(
    '#type' => 'submit',
    '#value' => t('Submit'),
  );
  $form['#submit'][] = 'my_form_add_more_submit';

  return $form;
}

function create_form(&$form, $row, $default_values = array()) {
  $form['agencies'][$row] = array('#type' => 'fieldset');
  $form['agencies'][$row]['component_name'] = array(
    '#type' => 'textfield',
    '#title' => t('Component name'),
    '#parent' => 'test',
    '#default_value' => isset($default_values['component_name']) ? $default_values['component_name'] : '',
  );
  $form['agencies'][$row]['agency_name'] = array(
    '#type' => 'select',
    '#title' => t('Select'),
    '#parent' => 'test',
    '#options' => array('1' => 'One', '2' => 'Two', '3' => 'Three'),
  );
  $form['agencies'][$row]['agency_id'] = array(
    '#type' => 'hidden',
    '#default_value' => isset($default_values['agency_id']) ? $default_values['agency_id'] : '',
  );
  $form['agencies'][$row]['agency_rate'] = array(
    '#type' => 'textfield',
    '#title' => t('Rate'),
    '#parent' => 'test',
    '#default_value' => isset($default_values['agency_rate']) ? $default_values['agency_rate'] : '',
  );
}

function ajax_simplest_callback($form, &$form_state) {
  return $form['agencies'];
}

function my_form_add_more_submit($form, &$form_state) {
  dpm($form_state['input']);
  dpm($form_state['values']);
}

Copy and paste the above code and you'll be able to test and learn how it works.  I don't like to give long explanations on how each part functions since most of that is located on the internet. This post is for seasoned Drupal programmers that need a quick answer.

NOTE: dpm() is used in my_form_add_more_submit(). You can download the devel module here, which includes dpm() functions.