what is the longest attested sequence of =ak that you've seen in Sumerian texts?
Hmm! I did a little searching, and here's what I've got.
"=ak" is one rendering of the Sumerian genitive case ending, roughly corresponding to English "of" or the possessive "apostrophe-S". It links two nouns together, X-Y-(ak) being "X of Y". The -a sound appears only after a consonant, and the -k- element only before a vowel and not at the end of a word, so in some circumstances (end of a word, after a vowel) it disappears entirely: lugal-uru "the king (lugal) of the city (uru)".
You can have multiple nested genitives, in which case the same rules apply: lugal-uru-e-k-a "the king of (-a) the city of (-k-) houses (e)", with the first -k- being the "inner" genitive and the second -a being the "outer" one. Jagersma (2010) lists several examples of this "stacked genitive", like tug-gu-anshe-k-a "the cloth of the neck of the donkey", as well as instances where additional case endings can be added after the genitive(s), like igi-shabra-Enlilakash "before (-sh) the eyes (igi) of (-a-) the administrator (shabra) of (-ak-) Enlil" - the final -akash represents three case endings in a row.
In theory, genitive elements can be stacked as long as you want, just as in English: "the power of the throne of the king of the city of the houses of copper" (or "the copper houses' city's king's throne's power", if you prefer). However, Edzard (2003) makes clear that "Probably for reasons of euphony, no more than two genitive particles were allowed on a string, even if three or more genitive relationships are involved." Jagersma gives an example of mashdaria-dam-shangnga-Ebarbara-k-ak-am "This is (-am) the ceremonial gift of the wife of the administrator of the Ebabbar temple," with three "of"s being covered by the two genitives in -k-ak-. Jagersma doesn't give an example with more than three nouns linked by these two genitives, but in theory you could have as many as you want. So the example at the start of this paragraph would be a-ashte-lugal-uru-e-urud-ak-a, with just the final two "-ak-a" representing all of the genitives - not *a-ashte-lugal-uru-e-urud-ak-ak-ak-ak-a, which would of course be ridiculous (^_~)
So if the question is "longest sequence of genitives", then I can find examples of three, but wouldn't doubt there are longer ones out there, and can construct my own of any length. But if the question is "longest sequence of =ak", the answer is... two. I hope that answers your question!