This is an odd case because it breaks the normal rules.
The normal English rule for possessives is: If the word is singular OR if it is a plural that does not end with "s", add apostrophe-s. If it is a plural that ends with "s", just add an apostrophe.
"one dog's bone" singular
"two dogs' bones" plural that ends in "s"
"one man's shirts" singular
"two men's shirts" plural that doesn't end in "s"
But here's where this case is odd. Normally we add the apostrophe-s or just apostrophe after a noun. But "else" is not a noun: it's an adjective. But it's an unusual adjective in that it comes after the noun. Usually we put adjectives before the noun. So when you make it possessive, in this case you put the apostrophe-s after an adjective. In such cases, you still indicate plural by making the NOUN plural, don't try to make the adjective plural. ("Anyone" is singular. I'm not sure that there's a plural of "anyone". "Anymany"?)
There are a few other examples in English where an adjective follows the noun. For example, in "attorney general" the noun is "attorney" and "general" is an adjective. That's "general" as in "not specific", not a military rank. So:
"one attorney general", "two attorneys general" To make it plural, add the "s" to the noun, not the adjective.
"one attorney general's office" Add the apostrophe-s to the end of the phrase, i.e. after the adjective.
"two attorneys general's offices" Add the "s" for plural to the noun. Then add an apostrophe-s for possessive to the end of the phrase.
Note: Many people make phrases like this plural by pluralizing the adjective, e.g. many make "mother-in-law" plural as "mother-in-laws". I'd say this is just wrong, but if you take the position that common usage is by definition correct, then this is probably well on its way to becoming correct. This is probably the biggest problem with the adjective "general", like "attorney general" and "consul general", because "general" can also be a noun and I suspect many people think a "consul general" is a general of type consul, rather than a consul of type general.
So "anyone else", "anyone else's".
Okay, long answer for a short question, but that's why they pay me the big bucks. Oh, wait ...