How can I add another option to this custom post function?

You just need to add a static <option> element before you start your loop.

<select style="width: 100px" name="games_database_rating">
  <option value="none">No Rating</option>
  <?php // Generate all items of drop-down list
      for ( $rating = 10; $rating >= 1; $rating -- ) {
  ?>
     <option value="<?php echo $rating; ?>" <?php echo selected( $rating, $game_rating ); ?>>
     <?php echo $rating; ?>/10 <?php } ?>
</select>

Now your select will start out with “No Rating”, and you can still choose a number. When you submit the form, just check for “none” or the number.