"Buffalo" is a solution to the following problem:

What English word can be repeated an arbitrary number of times to form a grammatically and semantically meaningful English sentence?

Buffalo has two meanings which are sufficient to solve the problem (though Webster 1913 only lists one: first, the common noun, meaning a large mammal, and second, a transitive verb, meaning to intimidate or to coerce. For the sake of clarity, in the following example sentences I will substitue the word bison for the animal and use buffalo exclusively as a verb.

This is a proof by induction. I will give examples of setences of length 1-3 and then give a description of how to take a sentence of length >= 3 (where length is an odd number) and form sentences of lengths length + 1 and length + 2

Length 1: "Buffalo!" is an imperative exclamation using the infinitive form of the verb.

Length 2: "Bison buffalo." is an existential statement that asserts that large four-legged mammals intimidate or coerce. (note that I will not promise that all of these sentences are, in fact, true, only that they are intelligible.

Length 3: "Bison buffalo bison." is an extension of our length 2 sentence that quantifies who the recipient of the coersion or intimidation is.

Length 4: "Bison bison buffalo buffalo." is the trickiest sentence to interpret. "Bison bison buffalo" is a noun phrase meaning, roughly, "The animals who are intimidated by other animals", and our sentence asserts that those intimidated animals are also guilty of intimidating (though just who they intimidate is not clear).

Length 5: "Bison bison buffalo buffalo bison." is the natural extension of our length 4 sentence, adding a direct object to the verb we added at the end of the last sentence.

So now we can create longer sentences by moving the direct object to the beginning of the sentence, creating a noun phrase ("The bison whom bison who bison buffalo buffalo...") and adding a verb to the end to make a sentence of length + 1, or adding a verb and a direct object to make a sentence of length + 2.

Please don't blame me if you get some strange stares from people when you tell them that buffalo buffalo buffalo buffalo buffalo.

Note: see also this node. There is some redundancy between that node and this one.