I came across the description of one exception on babbel.com:
One of the only exceptions in common use came into being in the late 1960s, and that is substituting the pronoun “sie,” which means she, for “es,” which means it, when discussing female children. The difficulty arose from the German rule which classifies all German nouns ending in “-lein” and “-chen,” both of which are diminutives, as neuter. Including the neuter definite article, the German noun for young girl is “das Mädchen”; therefore, German language authorities enacted the change so that people could address young girls in the third person using she.
The above paragraph seems to suggest that there are a few other rarer exceptions. What are these?
Some of the other exceptions that I know of are:
1) Some German nouns do not have a gender (e.g. Jura, meaning "Rechtswissenschaft als Studienfach"; Sanitär). I was asking a question about this earlier. This is not an official exception, but de facto, it is.
2) There is a question on German Stack Exchange that asks if “des Nachts” is the correct Genitive of the noun “die Nacht”. Ultimately, it turns out that it actually is. Notes in dict.cc indicate that the use of "des Nachts" is gehoben or veraltet. It would be very interesting to learn of other nouns that inflect in certain cases (Akk.; Dat. Gen.) as if they were of a different gender.
3) Mark Twain notes in his 1880 book A Tramp Abroad that the phrase “wegen des Regens” is correct. He adds, however, the following:
N. B. -- I was informed, later, by a higher authority, that there was an "exception" which permits one to say "wegen dem Regen" in certain peculiar and complex circumstances, but that this exception is not extended to anything but rain.
According to dict.cc, using Dative nowadays after wegen is not an exception, but rather a colloquial form of expression. Again, it would be very interesting to learn of a real non-colloquial exception, similar to the one above, valid in our days.
4) As can be seen from an answer to this question of mine, the preposition von is not always followed by a noun in the dative case. As it turns out, in some cases it can be followed by a noun in the nominative case!
5) Of all the letters in the German alphabet, only one exists exclusively in the lower-case-letter version: ß
6) The Hague (a city in the Netherlands) in English = Den Haag in German. As far as I know, city-names in German are either masculine or neuter, used in the nominative case. But Den Haag is used in the accusative case.