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Many (all?) languages have possession of some kind. Whether the language is possessor-possessum or possessum-possesor, the possessum is the head of the phrase.

I'm wondering whether the numeral or the noun is an expression like three chickens is the head and whether it varies from language to language.

In particular, is there a language without numeral classifiers where the numeral is clearly the head and a language where the noun is clearly the head?

Here's an excerpt from the WALS chapter on the order of numeral and noun.

In such languages [like Nusu, a Tibeto-Burman language with numeral classifiers], the numeral is not really modifying the noun, at least not directly, but rather is modifying the classifier. For such languages, the map shows the order of the noun relative to this phrase consisting of numeral and classifier. Whether this phrase actually modifies the noun is itself something that is often unclear. It has been proposed for some languages that the classifier is the real head of the noun phrase and the numeral and noun are both dependents of the classifier. See Map 55A on numeral classifiers.

The issue of what element is the head also sometimes arises sometimes in languages without numeral classifiers. It has been proposed that some numerals in Russian serve as heads (Babby 1987). Similarly, in Rif Berber (Morocco), most modifiers follow the noun, while numerals precede; but the construction they occur in is the same as the genitive construction, as illustrated in (12), suggesting that the numeral is the head, not a modifier.

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Yes, I think that it not only varies between languages, but it can vary within a language for different numerals. I think large numerals have a greater tendency to function as heads.

I think that the numeral is usually taken to be the head in English constructions like "hundreds/thousands/millions of beans", where the noun "beans" occurs inside of a prepositional phrase.

In other languages, there may be constructions where the noun takes a case that seems to function similarly to an "of" prepositional phrase in English. In Russian, I think the genitive case is involved, but it's a complicated topic that I don't know enough about to summarize.

English expressions like "three chickens" are tricky. In some ways, English numerals function similarly to articles. The indefinite article a(n) developed diachronically from one, and I've even seen some papers where it is argued that a(n) should be analyzed synchronically as a reduced form of the numeral one.* The definite article the is not derived from a numeral, but both numerals and articles can be categorized as taking a "determiner" function in a noun phrase.

The question of whether determiners or their associated nouns are heads (if determiners are the heads, it would technically be of a "determiner phrase" rather than of a "noun phrase") is a disputed topic, according to the Wikipedia article "Determiner phrase".

The section "Morphological dependencies" in that Wikipedia article mentions (although it doesn't provide any citations) that agreement with a noun can be taken to support an NP analysis over a DP analysis. English numerals don't show any agreement (although the determiners that/those and this/these do). But in other languages, a numeral may be a target for agreement in features like gender or case (or even, oddly enough, number: in Latin, the word for one, unus, has plural forms used with plural-only nouns). I think in such languages, it is usually assumed that the noun, which has the role of the controller, is the head.

Another complication in English: numerals can occur after determiners, which is a context where they typically could not be analyzed as having a determiner function. We can say "the three chickens", even though we can't say *"the my chickens"; one way of explaining that is to say that before a noun, my always functions as a determiner, but three doesn't always function as a determiner. But as with the NP vs. DP issue, I believe there are different approaches to analyzing the structure of expressions like "the three chickens". Some related posts: How to analyze an NP with two determiners? (Linguistics SE), One noun but two determiners? (ELU). According to Araucaria's answer to the second question, three in a phrase like "the three chickens" would function as a modifier.

*Here is a relevant passage from "On the Article in English", by David M. Perlmutter (1970):

If this is correct, the relationship between the definite article and the indefinite article in English is quite different from what has been generally supposed. Grammarians have worked on the assumption that NP's may bear either a definite or an indefinite article, and that the two constitute some sort of opposition. If the analysis given here is correct, however, the indefinite article is simply a numeral like all other numerals, and the occurrence or non-occurrence of the definite article is a completely independent phenomenon.

Further reading

Edit: here are some articles related to this question that I found after writing this post, but that I haven't finished reading yet. If anyone gets around to reading them before me and finds more relevant information to put into an answer, please go ahead and make a post of your own.

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  • It’s quite common for the number one to have a plural form – it does in English as well (as in “the green ones”, which is at least semantically different from the plurals of other numbers, like “three twos” in card games). Some languages have singular and plural forms of all numbers, like Finnish. Numerals may even generally agree with nouns in number. In Irish or Finnish, for example, numbers are generally used with the singular of the noun, because the number itself is singular; if the plural of the number is used, the plural of the noun is generally also used.
    – Janus Bahs Jacquet
    Commented Jul 18, 2019 at 19:27
  • (The only exception is, rather bizarrely, the Irish number one, which has no plural form itself, but is still used with a plural noun in the sense ‘any at all’. It’s always fascinated me that one can take a plural and two takes a singular in Irish.)
    – Janus Bahs Jacquet
    Commented Jul 18, 2019 at 19:29
  • German -en as a plural adjective inflectional morpheme, die grünen reflects PGem *-anaz IIRC. Adjectives can be nominalized simply by omiting the noun die grünen Politiker, die Grünen, and family names can be genitive pluralized (?) as "Simpsons". Thus, The ones may be simply falling in the same scheme. Something I had trouble parsing as a L2 was youngens, first heared in Al Bundy's big uns. Lulz
    – vectory
    Commented Jul 20, 2019 at 19:48
  • I saw an essay about the Three Stooges that began, “I always wanted to be a Three Stooge.” Hm, I thought, is there a language in which the core meaning of three is ‘belonging to a trio’?
    – Anton Sherwood
    Commented Aug 4, 2019 at 19:12

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