Drupal - Unlimited TextFields with Form API

This is not a form API level feature, it's field/widget API.

For a custom form, you have to build it yourself with an add more button that uses #ajax to add more elements. A full code example would be too complicated for an answer, but you have to keep track of the count in $form_state, on pressing of the button increaes it, rebuild the form and then return the new elements.

See https://api.drupal.org/api/drupal/core%21core.api.php/group/ajax/8.2.x for more documentation, a quick tutorial that I found: https://www.sitepoint.com/using-ajax-forms-drupal-8/

Here's the example from the examples module found here

use Drupal\Core\Form\FormStateInterface;

/**
 * Implements the ajax demo form controller.
 *
 * This example demonstrates using ajax callbacks to add people's names to a
 * list of picnic attendees.
 *
 * @see \Drupal\Core\Form\FormBase
 * @see \Drupal\Core\Form\ConfigFormBase
 */
class AjaxAddMore extends DemoBase {

  /**
   * Form with 'add more' and 'remove' buttons.
   *
   * This example shows a button to "add more" - add another textfield, and
   * the corresponding "remove" button.
   */
  public function buildForm(array $form, FormStateInterface $form_state) {
    $form['description'] = [
      '#type' => 'item',
      '#markup' => $this->t('This example shows an add-more and a remove-last button.'),
    ];

    // Gather the number of names in the form already.
    $num_names = $form_state->get('num_names');
    // We have to ensure that there is at least one name field.
    if ($num_names === NULL) {
      $name_field = $form_state->set('num_names', 1);
      $num_names = 1;
    }

    $form['#tree'] = TRUE;
    $form['names_fieldset'] = [
      '#type' => 'fieldset',
      '#title' => $this->t('People coming to picnic'),
      '#prefix' => '<div id="names-fieldset-wrapper">',
      '#suffix' => '</div>',
    ];

    for ($i = 0; $i < $num_names; $i++) {
      $form['names_fieldset']['name'][$i] = [
        '#type' => 'textfield',
        '#title' => $this->t('Name'),
      ];
    }

    $form['names_fieldset']['actions'] = [
      '#type' => 'actions',
    ];
    $form['names_fieldset']['actions']['add_name'] = [
      '#type' => 'submit',
      '#value' => $this->t('Add one more'),
      '#submit' => ['::addOne'],
      '#ajax' => [
        'callback' => '::addmoreCallback',
        'wrapper' => 'names-fieldset-wrapper',
      ],
    ];
    // If there is more than one name, add the remove button.
    if ($num_names > 1) {
      $form['names_fieldset']['actions']['remove_name'] = [
        '#type' => 'submit',
        '#value' => $this->t('Remove one'),
        '#submit' => ['::removeCallback'],
        '#ajax' => [
          'callback' => '::addmoreCallback',
          'wrapper' => 'names-fieldset-wrapper',
        ],
      ];
    }
    $form['actions']['submit'] = [
      '#type' => 'submit',
      '#value' => $this->t('Submit'),
    ];

    return $form;
  }

  /**
   * {@inheritdoc}
   */
  public function getFormId() {
    return 'form_api_example_ajax_addmore';
  }

  /**
   * Callback for both ajax-enabled buttons.
   *
   * Selects and returns the fieldset with the names in it.
   */
  public function addmoreCallback(array &$form, FormStateInterface $form_state) {
    return $form['names_fieldset'];
  }

  /**
   * Submit handler for the "add-one-more" button.
   *
   * Increments the max counter and causes a rebuild.
   */
  public function addOne(array &$form, FormStateInterface $form_state) {
    $name_field = $form_state->get('num_names');
    $add_button = $name_field + 1;
    $form_state->set('num_names', $add_button);
    // Since our buildForm() method relies on the value of 'num_names' to
    // generate 'name' form elements, we have to tell the form to rebuild. If we
    // don't do this, the form builder will not call buildForm().
    $form_state->setRebuild();
  }

  /**
   * Submit handler for the "remove one" button.
   *
   * Decrements the max counter and causes a form rebuild.
   */
  public function removeCallback(array &$form, FormStateInterface $form_state) {
    $name_field = $form_state->get('num_names');
    if ($name_field > 1) {
      $remove_button = $name_field - 1;
      $form_state->set('num_names', $remove_button);
    }
    // Since our buildForm() method relies on the value of 'num_names' to
    // generate 'name' form elements, we have to tell the form to rebuild. If we
    // don't do this, the form builder will not call buildForm().
    $form_state->setRebuild();
  }

  /**
   * Final submit handler.
   *
   * Reports what values were finally set.
   */
  public function submitForm(array &$form, FormStateInterface $form_state) {
    $values = $form_state->getValue(['names_fieldset', 'name']);

    $output = $this->t('These people are coming to the picnic: @names', [
      '@names' => implode(', ', $values),
    ]
    );
    $this->messenger()->addMessage($output);
  }

}