People in many regions of China may pronounce the same characters so differently that they can communicate with each other only by writing and not verbally. How do we define or measure mutual intelligibility in writing only (MIW hereinafter) in general? Here I describe an experiment that may serve as a starting point. Two people with high school or more education who natively speak language varieties A and B, respectively, but not both (if A and B are different), are subject to a test. Each person reads 100 sentences in normal speed randomly selected from the entire corpus of Modern A or B. (As an approximation to the entire corpus, take the Internet and book content indexed by Google as an example.) Each sentence is followed by 10 interpretations given in the language variety the other person understands, and he (she) chooses the correct one (10 choices instead of 4 or 5 just to reduce the random guess correctness). Then repeat the test switching the two people along with their respective language variety. If >=50 sentences are correctly understood, A and B are excluded from MIW. If it's <50, they are further subject to a test in which 100 sentences selected from the entire corpus are shown in writing. If >=90 sentences are correctly understood, we consider varieties A-B a case of MIW.
Thus, Sichuanese-Mandarin will be disqualified because they can be verbally communicated (and of course with written script). But Shanghainese-Mandarin, Hunanese-Mandarin, Shanghainese-Hunanese are good examples of MIW. Cantonese warrants more discussions. It's obvious that the Cantonese-Mandarin (or -Shanghainese etc.) pair has no verbal MI. There are grammar particles, pronouns and some common words unique to Cantonese. When a literate person who natively speaks Cantonese but has not learned the written Chinese in the way Chinese is taught in mainland China writes in Cantonese, can the writing be understood with >=90 correctness by one speaking Mandarin only? Suppose the content is absolutely randomly selected from the entire Cantonese corpus, and is not purely colloquial and definitely not contrived to contain a disproportionately high ratio of Cantonese-specific markers or characters. I don't know the answer, and an actual experiment is needed. One example of such a written script in Cantonese is a Wikipedia page. I personally don't know Cantonese and I may or may not be able to correctly answer 90 out of 100 questions in a reading comprehension test. Note that Cantonese is special in that many native Cantonese speakers do read Chinese text proficiently, although mandarin or other Chinese dialect speakers don't read Cantonese text (such as that Wikipedia page), creating asymmetric intelligibility, which is quite common in the world. Thus, when these two people try to communicate by writing, the preferred script they choose will be Chinese, not Cantonese. In discussing MIW, we should define two levels, one only allowing the written script to be the textual representation of the spoken language (e.g. Cantonese text for Cantonese speech), the other allowing the two people to choose whatever their preferred script is. Technically, we should limit MIW to the first case.
According to Wikipedia, Icelandic-Faroese and German-Dutch are MIW pairs. (The article also lists French-some Romance languages but does not give a good reference to support it. To my knowledge of a few Romance languages, this pair is invalid.) Based on a posting in Facebook Linguistics group, the following are additional language varieties that are candidates for MIW:
* Scots-English, and many languages in south Asia with Sanskrit roots
* Swiss German-Standard German
* Hanoi Vietnamese-Southern Vietnamese (but highly disputed in the Sinosphere group whose members are mostly Vietnamese)
* Danish-some other Scandinavian language
Note that I'm dealing with MIW between language varieties, a concept encompassing dialects within a language as well as languages, styles, registers, etc. While MIW within a language may not be limited to Chinese, it's probably safe to say the Chinese language has the most MIW pairs among its dialects, due to the dissociation between the pronunciation and the written form.