Consider a repository of citation information. containing books, articles, journals, edited books, and conference papers. Your task is to format the contents of the repository for consumption.
Let's take two approaches to formatting. National Library of Medicine (derived from the Vancouver Project) specifies citations in a particular way, mostly affecting how author names are laid out. NLM differs from American Psychological Association (APA) formatting. Your task is to provide a general approach to formatting citations using at least these two formatting standards. The visitor pattern is a good fit.
Now, you have to publish these citations and your choice of outputs are: plain text, PDF, HTML, RTF and MS Word (OOXML) or ODF. Some of these items require different layout strategies, depending on the type of the format (APA indents following lines, NLM doesn't).
"Double visitor" doesn't work so well. You really need something equivalent to "triple dispatch".
Now, that is not to say that there aren't approaches that get around the triple dispatch. For the above example, I ended up creating mixins and helper classes to provide the final publication version. These mixins and helpers are really where the 3rd dispatch occurred. I also evaluated publishing in a common denominator output and post-processing it to get the right output format, but that was ... bulky.