Install the Select or Other module
You can use the Select or Other module to add a dropdown with an Other option.
Selecting Other prompts the user to enter the new value in a separate field.
Your question asked to add the new business type directly; this module adds it to a separate field (however, an admin can later add the business type as a taxonomy term if you wish).
Advantages of this approach
- You get admin approval of user input. This way a bad user can't enter something like Sex sex sex which then appears as a business type on your site.
- You have control over how long the list gets instead of letting users add an unlimited number of terms.
Coding it yourself
You can also code this yourself pretty easily using the States API instead of installing the Select or Other module.
Add a new text field,
field_businesstype_other, to store the Other values for
Create a taxonomy term named Other in your "business type" vocabulary.
hook_form_alter(), add something like this:
$form['field_businesstype_other']['#states'] = array(
'visible' => array(
'select[name="field_businesstype"]' => array(
array('value' => '123'),
123 with the term ID of the Other term you created.