Update: Blue Magister's answer helped me to recognize a completely different use of the hypothetical its' than what I had been considering: as a predicate adjective.
I had been thinking of words like my and your—attributive adjectives—without considering the predicate forms mine and yours. So then I thought: "Maybe its' could be the predicate adjective form of the possessive pronoun for it?" Like...
You go your way, and I will go mine.
He went his way, and the horse went its'. (?)
Well, Dictionary.com put those insane notions to rest:
While it is possible to use its as a predicate adjective (The cat is angry because the bowl you're eating out of is its!) or as a pronoun meaning “that or those belonging to it” (Your notebook pages are torn. Borrow my notebook—its aren't), such use is rare and in most circumstances strained.
So, even in these extremely awkward cases, the correct word is still its and not its', at least according to Dictionary.com.
I just had a strange flashback to a conversation I had when I was in high school, with a man who was regarded by many members of a particular online community as having an impressive degree of knowledge of the English language.
The conversation centered on a claim this man made that I found very difficult to accept. I had made some remark involving the difference between it's and its (a distinction which I trust is quite well-known to the majority of users on this site), to which he had contributed, mostly phrased as an amusing aside, that there was one more word I hadn't mentioned: its', with the apostrophe at the end.
I originally thought he might have been joking, but we ended up debating this rather fervently. I seem to recall that I kept demanding he use the word in an example sentence, but he either could not or refused to do so. Yet he maintained that it is a word.
Is this true? I must concede I haven't put a lot of thought into it just now; but at the time, I was perplexed by the very suggestion that it could be a word (what could it mean?), and at the moment I can't really think of any scenario where it would make any sense.