First, your single line doesn't work because you can't string two separate s/// together like that.   What you wrote was

    $directory =~ s<^/*></>
               .   </*$></>;
[download]

Turning on warnings would've helped here.

cchampion fixed his RE so I can't explain why it doesn't work, cause now it does.

What I've been able to make work is something that recognizes the _middle_ of your string, not either end.   This is a translation out of Perl Cookbook, 2ndEd.   (cargo-cult of a higher order?)

    $directory =~ 
        s< ^  /?            # match leading '/' if present
              (             # capture what is between ends
                [^/]*       # match anything not a '/'
                (?:         # might be a '/', 
                  (?! /$ )  # but don't allow a '/' at EOL
                  .         # okay, eat the '/'
                ) *
              )
              /? $          # match trailing '/'
         >
         </$1/>x;    # now surround the middle with '/'
[download]

Update:   I take too long typing!   And my solution only takes one '/' off each end.   I guess I took so long typing that "I forgot the question"   ;-)