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I believe it's not possible in the plain drop down field. But it is possible in our widgets. You can check the available widgets here:
If you would like to let your users to add another option in a drop down, that is possible using this widget: http://widgets.jotform.com/widget/user_contributed_dropdown.
Or you can also use conditional logic to show and hide the "other" text box.
1. I added the default drop down field and a text box for the other.
2. In my conditional logic, if they selected the option "Other", then I'll display the "Other Text Box".
You can see the two workarounds I mentioned above in this sample form I made: http://form.jotformpro.com/form/51414820400945?.
I hope that helps.