current blog" in WPMU terms means the blog that is actually shown in the moment. In other words the current blog context of the page that is viewed in that moment or the content of
global $blog_id. As in a normal wp install you would use site_url() to retreive the link of that blog. (Just pointing this out so you don't get confused by that wording later.)
I think what you are searching for is the url to the blog which belongs to that user. There is no direct function for this. The best way to get this information is to use get_blogs_of_user() to get a list of blogs a user is attached to. If your setup is configured in a way that it's only possible for a user to have one blog, you can just use the first element from that array.
$blogs = get_blogs_of_user( get_current_user_id() );
$link = get_site_url($blogs->userblog_id);
<li class="showblog"><a href="<?php echo $link; ?>">Show blog</a></li>