I know there are plenty of languages out there where possessive personal pronouns can and do agree with the head noun in number and case. But I was wondering if there were any where nouns also had to agreed when placed in the possessive/genitive case? If grammatical number is marked with a suffix, then that means that such nouns may need to take two endings in the possessive: one to indicate the number of the noun, and another to agree with the number of the head noun.
This is possible, and it is my understanding that this phenomenon is called "Suffixaufnahme". One of the most well-known examples seems to be Old Georgian.
"Suffixaufnahme", by Marcus Kracht, gives the following examples, among others:
qeli-ta mocikul-ta-ta hand-obl.pl apostle-obl.pl-obl.pl through the hands of the apostles c̣inamsrbol-n-i lašḳar-ta-n-i forerunner-pl-nom army-obl.pl-pl-nom the vanguard of the armies
A few other related or just similar-looking, but distinct phenomena are sometimes also called "Suffixaufnahme". I think Valdeut gives a good overview in the following post on the thread "Basque's Surdéclinaison" at the Zompist Bulletin Board:
in some languages, such as Sumerian, it seems the head noun usually comes at the start of the noun phrase, but case markers are applied at the end of the noun phrase rather than at the end of the head noun. The result is that embedding causes multiple case markers to appear after each other at the end of the noun phrase. Wikipedia cites the following example:
sipad udu siki-ak-ak-ene ("the shepherds of woolly sheep"), where the first genitive morpheme (-a(k)) subordinates siki "wool" to udu "sheep", and the second subordinates udu siki-a(k) "sheep of wool"
[...] Zólyomi, Gábor (2014). Grzegorek, Katarzyna; Borowska, Anna; Kirk, Allison, eds. Copular Clauses and Focus Marking in Sumerian. De Gruyter. p. 8. ISBN 978-3-11-040169-1. Retrieved 21 July 2016.
Valdeut mentions the term "Suffixhäufung" as a name for this phenomenon.
in some languages, such as Basque, there is not agreement within the noun clause, but an additional case marker may be added after a case marker in contexts of afterthought or when an inflected word is used by itself (with ellipsis of the head noun). Some people seem to avoid using "Suffixaufnahme" to refer to this. As the linked thread indicates, the French term "surdéclinaison" has been used to refer to the phenomenon seen in Basque.