I am completely new to linguistics (and have to write an essay linked to this topic) and have read about Merge, which seems to allow its outputs to be put back in as inputs. Would you say this shows how we can create infinite combinations of words? I remember reading some of Steven Pinker's work, and he was saying the fact that you can always add a prefix to a sentence (e.g. 'Roger said, "..."') means you can never have a longest sentence. I understand that, so I would like to know if this is essentially what Merge shows with recursion. Sorry if this whole question is incoherent, I am a total novice and need educating!
I think rather than saying that the recursive capability of Merge "demonstrate[s] the productivity of human language", it would make more sense to say that it explains the productivity that we see. The idea of "Merge" acts as a kind of explanation; I might be misunderstanding the wording of your question, but I don't think it makes sense to say that an explanation demonstrates or shows the phenomenon that it was introduced to explain.
My understanding is that linguists who believe in "Merge" think that it is responsible for the ability to create infinite combinations of words that is a feature of many well-known languages, such as English.
But the existence of "Merge" by itself doesn't establish that no human language could have a longest sentence. Clearly, languages use rules other than "Merge" to determine what sentences and syntactic structures are grammatical. For example, I believe some languages don't allow embedded genitive structures, like "Alice's brother's car".
The definition of "recursion" that is used in current Generativist literature seems to be equivalent to "computable", plus maybe some other details, although you shouldn't take my word for that as I don't understand computability theory. I wrote an answer about it here: https://linguistics.stackexchange.com/a/26206/5581 The brouhaha about the supposed absence of embedded clauses in Piraha seems to have inspired a fair number of responses from Generative Grammar types that seem to say that the existence of "Merge" isn't expected to imply that all particular human languages will allow embedding, or the construction of an infinite number of sentences of indefinite length. My understanding is that in current theory, "Merge" is also supposed to be used in putting together words of different categories; e.g. it is "recursive" in the sense that after applying "Merge" to "red" and "house" to get "red house", "Merge" is applied again to "the" and "red house" to get "the red house" and then again to "the red house" and "is on the corner" to get the sentence "the red house is on the corner". So while the capability of constructing sentences of indefinite length in some languages may be taken as evidence for the existence of "Merge", my impression is that contrived examples of the kind "Mary said "Roger said "Sally said ... " " " that show recursion in the sense of embedding phrases of the "same" category in one another don't seem to be such important examples as they are often made out to be. I don't know too much about the history of this area, though, so maybe there was some reason for choosing these examples.
There is more relevant info in the answers to the following question: What's the difference between recursion and embedding?
It seems pretty obvious that people don't just memorize a look-up table of sentences when they learn a language, but apparently there's some disagreement about how to describe and explain this fact.