[GuardCat]

<input type='checkbox' name='chec' id='chec' value='1000'>
    <input type='checkbox' name='chec'  value='100'>
    <input type='checkbox' name='chec' value='10'>
    <input type='checkbox' name='chec' value='1'>
    
    <input type = "button" id = "show" value = "show me">


    <script>
      function getCheckboxesValues( name ) {
        var x, result = "", checkboxes = document.getElementsByName( name );
        
        for ( x = 0; x < checkboxes.length; x++ ) {
          result += checkboxes[ x ].checked ? checkboxes[ x ].value : ""
        }
        
        return result
      }
      
      document.getElementById( "show" ).onclick = function() {
        alert( getCheckboxesValues( "chec" ) )
      }
    </script>

checkbox, а не chekbox
И одинаковые id делать не стоит.