Get the most recently modified post date of most recently modified post

Try this:

$q = new WP_Query;

$posts = $q->query( array(
    'posts_per_page' => 1,
    'orderby'        => 'post_modified', // Sorts by the date modified.
    'order'          => 'DESC',          // Sorts in descending order.
    'no_found_rows'  => true,
) );

// Note that $posts could have more than one post, if your site have sticky
// posts. However, the first post, unless filtered by plugins, would always
// be the most recently modified/updated one.
if ( ! empty( $posts ) ) {
    $post = $posts[0];
    setup_postdata( $post );
    ?>
        <p>Most recent site update: <?php the_modified_date( 'F d, Y' ); ?>.</p>
    <?php
    wp_reset_postdata();
}