There is a way to do what you want.
After creating new drop-down field, make ‘other’ entry as a last choice. Create another, blank ‘Text box’ field and call it whatever you want. For this example I’ll call it ‘Other’.
Now create a condition (see example http://www.jotformpro.com/form/22490977052963):
If ‘drop-down’ field ‘Not equals to’ ‘Other’
Then ‘Hide’ ‘Other’ Field
So, as the form loads, the ‘Other’ field will be hidden. If user selects ‘Other’ from a drop-down, the field will be shown. This can be done for more than one drop-down filed or for various options in the same field.