User Defined order on get_categories?

I think the most sane way and easiest way of doing this is to unset the first value in the returned array, and then adding it back at the end of the returned array before your foreach loop

For this to work, you will need to sort your categories by ID as you need to take category ID 1 and add that to the back. Something like this will work

<?php
$args = array(
    'hierarchical'             => 1,
    'orderby'                  => 'id',
    'order'                    => 'ASC',
    'include'                  => '13,21,41,1',
); 

$categories = get_categories( $args );

$v = $categories[0];
    unset($categories[0]);
    $categories[0] = $v;


foreach( $categories as $cat) {
    echo $cat->slug ;
}
?>