Arguments for, and the applicability of, using multiple forms of compression on one or more files?

I was wondering whether there are any scenarios which require file(s) to be compressed multiple times with the same or a different algorithm. As far as i can tell the file size can be reduced further by applying further compression dependent on the nuances of a compression algorithm. (what files or data it is optimised for). Perhaps certain technologies expect a certain file type? If you apply multiple rounds of compression could it result in an increasing file size, as information is added to denote that a certain compression encoding has been used?