-- HTML
Your score is <span>0</span>
<button>Click me</button>

-- Javascript include jquery
$('button').click(function() {
   $.get('file.php', function(data) {
     $('span').html(data);
   }
})

-- PHP
<?php
session_start();
if (!isset($_SESSION['scorekeeper']) {
  $_SESSION['scorekeeper'] = 1; // 1 because this is the first button press
} else {
  $_SESSION['scorekeeper']++;
}

echo $_SESSION['scorekeeper']