I still encounter this problem.

Excellent kata!

I have a small remark. The description is not precise.

There is a infinite string. You can imagine it's a combination of numbers from 1 to n, like this:

It is not just a combination. It is the infinite sequence of digits defined as the concatenation of the sequence of strings representing the consecutive numbers from 1 to infinity.

In the description, the reduce() method is required to return a lambda object, but the tests expect a string instead.

I'm also thinking about it (alpha-reduction). In general, the substitution operation requires that the variable "v" bound by an abstraction "lambda v.body" shouldn't be free in a term which is to replace some variable in the abstraction body
(one can say "v" should be "fresh" for the substitution term). Otherwise, an alpha-reduction should be done to avoid changes in semantics.
In some random tests such a situation may appear. In my solution https://www.codewars.com/kata/reviews/55f3256f0fd3072e02000115/groups/64f846e2717c5300019e476e, I throw an error then. Anyway, if I ignore it and "force" substitution with usual rules, all tests are passed.

It's a version before refactoring. I wonder why it passed the tests.

This one is valid.

Mine does not use JSON.stringify(). I'm comparing arguments by strict equality.

This kata in JavaScript can't be done using Cheerio only. I've done it with Puppeteer but, unfortunately, I wasn't allowed to import Puppeteer here in order to submit my solution.

A useful fact:
It is sufficient to check if there exist a left neutral element "e" (ie. e * g == g for all g) and each element "g" has a left inverse (ie. there is "h" such that h * g == e).
Then, under associativity and using a few tricky comparisons, we can prove that any left inverse is also a right inverse, "e" is (two-sided) neutral element, from which the uniqeness of "e" and all inverses follows.

Oh, I've just seen that number_of_moves is not necessary. It suffices to return len(history) - 1.
Nice kata!