Auto resizing the SELECT element according to selected OPTION's width

You are right there is no easy or built-in way to get the size of a particular select option. Here is a JsFiddle that does what you want.

It is okay with the latest versions of Chrome, Firefox, IE, Opera and Safari.

I have added a hidden select #width_tmp_select to compute the width of the visible select #resizing_select that we want to be auto resizing. We set the hidden select to have a single option, whose text is that of the currently-selected option of the visible select. Then we use the width of the hidden select as the width of the visible select. Note that using a hidden span instead of a hidden select works pretty well, but the width will be a little off as pointed out by sami-al-subhi in the comment below.

$(document).ready(function() {
  $('#resizing_select').change(function(){
    $("#width_tmp_option").html($('#resizing_select option:selected').text()); 
    $(this).width($("#width_tmp_select").width());  
  });
});

#resizing_select {
  width: 50px;
} 
 
#width_tmp_select{
  display : none;
} 

<script src="//cdnjs.cloudflare.com/ajax/libs/jquery/2.1.3/jquery.js"></script>

<select id="resizing_select">
  <option>All</option>
  <option>Longer</option>
  <option>A very very long string...</option>
</select>

<select id="width_tmp_select">
  <option id="width_tmp_option"></option>
</select>

Here's a plugin I just wrote for this question that dynamically creates and destroys a mock span so it doesn't clutter up your html. This helps separate concerns, lets you delegate that functionality, and allows for easy reuse across multiple elements.

Include this JS anywhere:

(function($, window){
  var arrowWidth = 30;

  $.fn.resizeselect = function(settings) {  
    return this.each(function() { 

      $(this).change(function(){
        var $this = $(this);

        // create test element
        var text = $this.find("option:selected").text();
        var $test = $("<span>").html(text).css({
            "font-size": $this.css("font-size"), // ensures same size text
            "visibility": "hidden"               // prevents FOUC
        });


        // add to parent, get width, and get out
        $test.appendTo($this.parent());
        var width = $test.width();
        $test.remove();

        // set select width
        $this.width(width + arrowWidth);

        // run on start
      }).change();

    });
  };

  // run by default
  $("select.resizeselect").resizeselect();                       

})(jQuery, window);

You can initialize the plugin in one of two ways:

1. HTML - Add the class .resizeselect to any select element:

   <select class="btn btn-select resizeselect">
     <option>All</option>
     <option>Longer</option>
     <option>A very very long string...</option>
   </select>
   

2. JavaScript - Call .resizeselect() on any jQuery object:

   $("select").resizeselect()
   

Here's a Demo in jsFiddle

Here's a Demo in Stack Snippets:

(function($, window){
  var arrowWidth = 30;

  $.fn.resizeselect = function(settings) {  
    return this.each(function() { 

      $(this).change(function(){
        var $this = $(this);

        // create test element
        var text = $this.find("option:selected").text();
        var $test = $("<span>").html(text).css({
        	"font-size": $this.css("font-size"), // ensures same size text
            "visibility": "hidden"               // prevents FOUC
        });
        
        // add to parent, get width, and get out
        $test.appendTo($this.parent());
        var width = $test.width();
        $test.remove();

        // set select width
        $this.width(width + arrowWidth);

        // run on start
      }).change();

    });
  };

  // run by default
  $("select.resizeselect").resizeselect();                       

})(jQuery, window);

<script src="//cdnjs.cloudflare.com/ajax/libs/jquery/2.1.3/jquery.js"></script>

<select class="resizeselect">
  <option>All</option>
  <option>Longer</option>
  <option>A very very long string...</option>
</select>

Updated to include sizing suggestions from Garywoo & Eric Warnke

This working solution makes use here of a temporary auxiliary select into which the selected option from the main select is cloned, such that one can assess the true width which the main select should have.

The nice thing here is that you just add this code and it's applicable to every selects, thus no need to ids and extra naming.

$('select').change(function(){
  var text = $(this).find('option:selected').text()
  var $aux = $('<select/>').append($('<option/>').text(text))
  $(this).after($aux)
  $(this).width($aux.width())
  $aux.remove()
}).change()

<script src="https://cdnjs.cloudflare.com/ajax/libs/jquery/3.3.1/jquery.min.js"></script>
<select>
  <option>REALLY LONG TEXT, REALLY LONG TEXT, REALLY LONG TEXT</option>
  <option>ABCDEFGHIJKL</option>
  <option>ABC</option>
</select>