A dropdown control is useless if it is not bound to a field. If there is not a field to store the control's value, the control can have no value.
But what you can do if you really don't want to store the value in the form's main data source is to create a simple "Form Logic" secondary data source from an .XML file. It can be something like this:
<?xml version="1.0">
<formLogic>
<!-- Pick a more meaningful name and initial value for this next field -->
<dropdownValue>widgets</dropdownValue>
</formLogic>
One consequence of this, though, is that the value of the dropdown will not persist if you close and re-open the form.