another kind of infinite compression

In another thread, Phil Carmody recently talked about the
theoretical maximum compression ratio of zlib and about
recompressing gzip data.  This reminded me of a question I've
had for a while.  The short version is:  is it possible to have
a compressed archive file that contains itself and some other
data as well?

For example, is there any foo.tar.gz that I could extract and
get a copy of the same foo.tar.gz plus some other file(s)?

I have thought about this a little, and I'm pretty confident
that it is possible in a theoretical sense.  What I'd like to
know is whether it's possible with any commonly-used existing
archive format (.tar.gz, .tar.bz2, .tar.Z, .zip, .sit, etc.)
or any other archive format not specifically designed to make
it happen.  :-)

I would not care if the filename changes or if it takes several
rounds of extraction to get the original file back.  (I believe
allowing several rounds of extraction makes the whole thing
equivalent to asking whether there is any foo.tar.gz that, when
recursively extracted, produces an infinite amount of data.)

  - Logan