There is no simple answer to this. In fact, there may be no standard answer. Language use, including punctuation changes over time.
"Each other's" (or "each others' ") is usually followed by a noun. And most people might say that "each other's book" is correct for a count noun such as book. But others will insist that "each other's books" is correct. Because more than one book is involved. In addition, many, many authors (I mean authors, not students) use
"Each others' books." Some people, so confused about the issue eliminate the apostrophe (') and just write "each others book" or "each others books." The apostrophe and the period are going out of style in some usages, for instance, we used to always abbreviate "mister" as Mr. and "doctor" as Dr., but now we usually can spell it as Mr and Dr, without the period.
Anyway, right now, most people will say to use:
"each other's books" for a count noun, or
"each other's rice" for a non-count noun.
But you will find "each others' books" used often. As well as "each others' books." I say take your pick. But I am not a grammar policeman, like many people/teachers are. The main reason in the case of this phrase is because in spoken English both "each other's" and "each others' " sound the same.
Note, however, that I don't believe I have ever heard, or written, or read
each other's' until I read petrifyer's answer.