The essence of human language is, according to Chomsky, the ability of finite brains to produce what he considers to be infinite grammars. By this he means not only that there is no upper limit on what we can say, but that there is no upper limit on the number of sentences our language has, there's no upper limit on the size of any particular sentence. Chomsky has claimed that the fundamental tool that underlies all of this creativity of human language is recursion: the ability for one phrase to reoccur inside another phrase of the same type. If I say "John's brother's house", I have a noun, "house", which occurs in a noun phrase, "brother's house", and that noun phrase occurs in another noun phrase, "John's brother's house". This makes a lot of sense, and it's an interesting property of human language.
On this level, we have the last output of "John's" being re-entered into the meaning, so that we know it isn't John we're talking about, but his brother's house.
Let's contrast this to what the Piraha would say:
So in the case of Pirahã, the language I've worked with the longest of the 24 languages I've worked with in the Amazon, for about 30 years, Pirahã doesn't have expressions like "John's brother's house". You can say "John's house", you can say "John's brother", but if you want to say "John's brother's house", you have to say "John has a brother. This brother has a house". They have to say it in separate sentences.
On the level of the sentence, there is no possibility of chaining possessives, but the overall meaning of one sentence is carried over into the next and in fact it would make no sense to say "this brother" without the preceding sentence. "This brother" in fact refers back (self-refers, calling itself...) and is therefore fundamentally recursive. Because the chunking system does not allow recursion within sentences, it must allow recursion between sentences. "This" is an earmark of recursion because it is infinitely various, it refers to "the last thing" and is still built on LISP-like linked list principles which are different from "Infinite Language" that "uses recursion" only insofar as the rules which allow it to be employed.
Recursion is not permitted on the level of the sentence, because the sentence, like a word in English, conveys a single thought. On some level, every language is finite. Containing a finite set of agreed upon meanings regardless of articulation. The finitude in fact allows us to define. Without it, everything would remain undefined.
The endlessness with which such a culture could string their sentences together shows me that the set of all things described in tandem is up to the author's attention to detail, and does not limit the possibility of what things can be described. It may refuse certain possibilities but (Infinity - Any number), does not mean that number is finite.
At any rate, my only knowledge of this culture is from the aforementioned article so I take it on faith that these people have violated one of Chomsky's laws. However, we can still say that recursion occurs within this language, so Chomsky's law can be relaxed without causing a cascade failure of Chomskyan linguistics. My only point in making this argument is to corrupt previously held ideas about finite-ness and recursion. The idea of referencing back is necessarily recursive, in fact "this" is a tell tale sign of recursion. But perhaps not the kind Chomsky is talking about.