Based on (this answer) WooCommerce Cart Quantity Base Discount, you can add a progressive fee based on total number of items:
add_action( 'woocommerce_cart_calculate_fees','woocommerce_cart_extra_cost', 10, 1 );
function woocommerce_cart_extra_cost( $cart_object ) {
if ( is_admin() && ! defined( 'DOING_AJAX' ) ) return;
$cart_item_count = $cart_object->get_cart_contents_count();
// CONDITIONAL ITEMS QUANTITY FEE AMOUNT
if( $cart_item_count < 6 )
$fee = 0;
elseif( $cart_item_count >= 6 && $cart_item_count < 12 )
$fee = 5;
elseif( $cart_item_count >= 12 )
$fee = 10;
if( $fee > 0 )
$cart_object->add_fee( __( "Extra Cost", "woocommerce" ), $fee, true);
}
Code goes in function.php file of your active child theme (or theme) or also in any plugin file.
Tested and works.