Custom form with multiple textarea fields How can I create a textarea field in a custom module which has multiple (unlimited) fields? It should be the same result as selecting unlimited amount of values when adding a textarea fields to a content-type.
There is no property in Drupal's Form API to create multi value fields. You have to create that functionality manually. Check this tutorial.

[Edit] the tutorial seems to have a couple errors (e.g. the values entered in names are not inherited as default_value when the form is rebuilt when adding a field). But it can give you a general idea on how do this.