Nested Shortcode via Filter

add-nested-shortcode.php

/**
 * This will allow you to grab the output of an existing shortcode (you're probably already using)
 * and execute the shortcodes you're using inside of it.
 */
function my_function_name ( $output, $tag ) {
	if ( 'existing_shortcode_name' != $tag ) {
		return $output;
	}
	$output = do_shortcode($output);
	return $output;
}
add_filter('do_shortcode_tag', 'my_function_name', 10, 2);