How to use update_option() without replacing/losing the old value?

Use update_option() in conjunction with get_option():

$old_option = get_option('blogdescription');
$new_option = 'My new blog description';
$new_option .= $old_option; // This is the same as $new_option = $new_option . $old_option;
update_option('blogdescription', $new_option);

If they are integers, obviously use something like $new_option += $old_option.