Please refrain from top-posting.

I trimmed your SSCCE, here's my version.
<sscce>
import java.awt.*;
import javax.swing.*;

public class test1 extends JFrame {
   public test1 () {
       super();
       setSize(200,200);
       setDefaultCloseOperation(
           JFrame.EXIT_ON_CLOSE);

       // do you recall me mentioning
       // 'content pane' earlier?
       Container contentPane = getContentPane();
       contentPane.setLayout(new FlowLayout());

       String[] list = {"one", "two", "three"};
       JComboBox box = new JComboBox(list);

       box.setEditable(true);
       contentPane.add(box);
       // do you recall me mentioning
       // 'validate' earlier?
       validate();
   }

   public static void main (String[] args) {
       test1 s = new test1();
       s.setVisible(true);
   }
}
</sscce>

The drop-down values can be selected in this version.

Andrew T.

                                 ^^^^
         Container rootPane = getContentPane();
           

This is the mistake.  You want to add the component to the Frame's
contentPane, not the rootPane.