Is [Its'] a word? (Note the apostrophe at the end.)

I found a Yon It living in Long Beach, CA. His (or her) family would be the Its. If they had a dog, it would be the Its' dog.

It is raining, isn't it?

There are two its in this sentence. The its' positions are the beginning and end.

That word does not exist. "Its" is a possesive pronoun, so it already means "belonging to".

Source here.

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.