You can do it as follows,
$('form input').on("propertychange keyup input paste", addInput);
    function addInput() {
        var remainingChars = $(this).val().length;
        if (remainingChars == 24) {
            $('<input type="text">').change(fileChangeHandler).appendTo('form').focus().on("propertychange keyup input paste",addInput);
        }}
http://jsfiddle.net/5KZtB/
EDIT - store input values
To store the values of the fields in an array you can push each value before creating the new field. For example,
jQuery(function($) {
    var values = [];
      $('form input[type=text]').change(fileChangeHandler);
      function fileChangeHandler() {
        var form = $(this).closest('form');        
      }
     var form = $(this).closest('form');
    $('form input').on("propertychange keyup input paste",
    addInput);
    function addInput() {
        var remainingChars = $(this).val().length;
        if (remainingChars == 24) {
            $('<input type="text">').change(fileChangeHandler).appendTo('form').focus().on("propertychange keyup input paste",addInput);
            values.push($(this).val());
            //console.log(values);
        }}
    });
http://jsfiddle.net/5KZtB/1/
EDIT2 - allow only numeric
....
    $('form input').on("propertychange keyup input paste",
    addInput);
    onlyNums($('form input'));
    function addInput() {
        var remainingChars = $(this).val().length;
        if (remainingChars == 24) {
            var newInput = $('<input type="text">').change(fileChangeHandler).appendTo('form').focus().on("propertychange keyup input paste",addInput);
            onlyNums(newInput);
            values.push($(this).val());
            console.log(values);
        }}
    function onlyNums($elem){
        $elem.keydown(function(event) {
        // Allow only backspace and delete
        if ( event.keyCode == 46 || event.keyCode == 8 ) {
            // let it happen, don't do anything
        }
        else {
            // Ensure that it is a number and stop the keypress
            if (event.keyCode < 48 || event.keyCode > 57 ) {
                event.preventDefault(); 
            }   
        }
    });
    }
http://jsfiddle.net/5KZtB/2/
EDIT3 - a field to delete the input elements
A rough approach could be to add the input inside a span element as well as a link with an x mark for deleting the specific input. So the code has been slightly modified as follows,
HTML
<form>
    <span class="input-component"><input type="text"/><a href=#>[x]</a></span>
  </form>
JS
jQuery(function($) {
    var values = [];
      $('form input[type=text]').change(fileChangeHandler);
      function fileChangeHandler() {
        var form = $(this).closest('form');        
      }
     var form = $(this).closest('form');
    $('form .input-component input').on("propertychange keyup input paste",
    addInput);
    onlyNums($('form .input-component input'));
    initializeDelLink($('form .input-component'));
    function addInput() {
        var remainingChars = $(this).val().length;
        if (remainingChars == 24) {
            var $newInputComponent = $('<span class="input-component"><input type="text"/><a href=#>[x]</a></span>').change(fileChangeHandler).appendTo('form').find('input').focus().on("propertychange keyup input paste",addInput).end();
            initializeDelLink($newInputComponent);
            var $newInput = $newInputComponent.find('input');
            onlyNums($newInput);
            values.push($(this).val());
            console.log(values);
        }}
    function initializeDelLink($elem){
        var $delLink=$elem.find('a');
            $delLink.click(function(){alert();
                $(this).parent('.input-component').remove();
            });
    }
    function onlyNums($elem){
        $elem.keydown(function(event) {
        // Allow only backspace and delete
        if ( event.keyCode == 46 || event.keyCode == 8 ) {
            // let it happen, don't do anything
        }
        else {
            // Ensure that it is a number and stop the keypress
            if (event.keyCode < 48 || event.keyCode > 57 ) {
                event.preventDefault(); 
            }   
        }
    });
    }
    });
http://jsfiddle.net/5KZtB/3/