In functions.php:

//total site post count
function post_count() {
    global $wpdb;
    if ( false === ( $cached_post_count = get_transient( 'cached_post_count' ) ) ) {
        // It wasn't there, so regenerate the data and save the transient
        $cached_post_count = $wpdb->get_var('SELECT COUNT(*) FROM ' . $wpdb->posts . ' WHERE post_status = "publish" AND post_type = "post"');
        set_transient( 'cached_post_count', $cached_post_count, 60*60*24 );
    }
    return $cached_post_count;
}

Then display it anywhere in your theme:

<?php echo get_transient(cached_post_count); ?>