Throughout the POSIX specification, there's provision (11, 22, 33...) to allow implementations to treat a path starting with two / specially.
A POSIX application (an application written to the POSIX specification to be portable to all POSIX compliant systems) cannot assume that //foo/bar is the same as /foo/bar (though they can assume that ///foo/bar is the same as /foo/bar).
Now what are those POSIX systems (historical and still maintained) that treat //foo specially? I believed (I've now been proven wrong) that POSIX provision was pushed by Microsoft for their Unix variant (XENIX) and possibly Windows POSIX layer (can anyone confirm that?).
It is used by Cygwin which also is a POSIX-like layer for Microsoft Windows. Are there any non-Microsoft Windows systems? OpenVMS?
On systems where //foo/bar is special, what is it used for? //host/path for network file systems access? Virtual file systems?
Do some applications running on Unix-likes —if not the system's API— treat //foo/bar paths specially (in contexts where they otherwise treat /foo/bar as the path on the filesystem)?
Edit, I've since asked a question on the austin-group mailing list about the origin of //foo/bar handling in the spec, and the discussion is an interesting read (from an archaeology point of view at least).