Drop Down Menu/Text Field in one

The modern solution is an input field of type "search"!

https://developer.mozilla.org/en-US/docs/Web/HTML/Element/input/search https://www.w3schools.com/tags/tag_datalist.asp

Somewhere in your HTML you define a datalist for later reference:

<datalist id="mylist">
   <option value="Option 1">
   <option value="Option 2">
   <option value="Option 3">
</datalist>

Then you can define your search input like this:

<input type="search" list="mylist">

Voilà. Very nice and easy.

Option 1

Include the script from dhtmlgoodies and initialize like this:

<input type="text" name="myText" value="Norway"
       selectBoxOptions="Canada;Denmark;Finland;Germany;Mexico"> 

createEditableSelect(document.forms[0].myText);

Option 2

Here's a custom solution which combines a <select> element and <input> element, styles them, and toggles back and forth via JavaScript

<div style="position:relative;width:200px;height:25px;border:0;padding:0;margin:0;">
  <select style="position:absolute;top:0px;left:0px;width:200px; height:25px;line-height:20px;margin:0;padding:0;"
          onchange="document.getElementById('displayValue').value=this.options[this.selectedIndex].text; document.getElementById('idValue').value=this.options[this.selectedIndex].value;">
    <option></option>
    <option value="one">one</option>
    <option value="two">two</option>
    <option value="three">three</option>
  </select>
  <input type="text" name="displayValue" id="displayValue" 
         placeholder="add/select a value" onfocus="this.select()"
         style="position:absolute;top:0px;left:0px;width:183px;width:180px\9;#width:180px;height:23px; height:21px\9;#height:18px;border:1px solid #556;"  >
  <input name="idValue" id="idValue" type="hidden">
</div>

I found this question and discussion very helpful and wanted to show the solution I ended up with. It is based on the answer given by @DevangRathod, but I used jQuery and made a couple tweaks to it, so wanted to show a fully commented sample to help anyone else working on something similar. I originally had been using the HTML5 data-list element, but was dissatisfied with that solution since it removes options from the drop down list that don't match text typed in the box. In my application, I wanted the full list to always be available.

Fully functional demo here: https://jsfiddle.net/abru77mm/

HTML:

<!-- 
Most style elements I left to the CSS file, but some are here.
Reason being that I am actually calculating my width dynamically
in my application so when I dynamically formulate this HTML, I
want the width and all the factors (such as padding and border
width and margin) that go into determining the proper widths to
be controlled by one piece of code, so those pieces are done in
the in-line style.  Otherwise I leave the styling to the CSS file.
-->
<div class="data-list-input" style="width:190px;">
  <select class="data-list-input" style="width:190px;">
    <option value="">&lt;Free Form Text&gt;</option>
    <option value="Male">Male</option>
    <option value="Female">Female</option>
    <!-- Note that though the select/option allows for a different
    display and internal value, there is no such distinction in the
    text box, so for all the options (except the "none" option) the
    value and content of the option should be identical. -->
  </select>
  <input class="data-list-input" style="width:160px;padding:4px 6px;border-width:1px;margin:0;" type="text" name="sex" required="required" value="">
</div>

JS:

jQuery(function() {
  //keep the focus on the input box so that the highlighting
  //I'm using doesn't give away the hidden select box to the user
  $('select.data-list-input').focus(function() {
    $(this).siblings('input.data-list-input').focus();
  });
  //when selecting from the select box, put the value in the input box
  $('select.data-list-input').change(function() {
    $(this).siblings('input.data-list-input').val($(this).val());
  });
  //When editing the input box, reset the select box setting to "free
  //form input". This is important to do so that you can reselect the
  //option you had selected if you want to.
  $('input.data-list-input').change(function() {
    $(this).siblings('select.data-list-input').val('');
  });
});

CSS:

div.data-list-input
{
    position: relative;
    height: 20px;
    display: inline-flex;
    padding: 5px 0 5px 0;
}
select.data-list-input
{
    position: absolute;
    top: 5px;
    left: 0px;
    height: 20px;
}
input.data-list-input
{
    position: absolute;
    top: 0px;
    left: 0px;
    height: 20px;
}

Any comments for improvement on my implementation welcome. Hope someone finds this helpful.

You can do this natively with HTML5 <datalist>:

<label>Choose a browser from this list:
<input list="browsers" name="myBrowser" /></label>
<datalist id="browsers">
  <option value="Chrome">
  <option value="Firefox">
  <option value="Internet Explorer">
  <option value="Opera">
  <option value="Safari">
  <option value="Microsoft Edge">
</datalist>