As far as I know, the LDS church is the only religion of note to have ever claimed that every can become gods. Not that everyone will, but that everyone can. If they haven't received the necessary ordinances in life, then there's still a chance in death through proxy ordinances.
This belief began with Joseph Smith. Its not taught very openly, and neither me or my brother learned of it until our teenage years despite being born into the church. Its a very important part of "The Plan of Salvation," yet Mormons don't usually tell prospective converts about that part. It's too...strange.
It came up between Steven and I within a few months of his conversion, though. I loved that part of Mormon doctrine, and thought it made so much sense. God made us in his image; he wants us to become like him. The son follows the footsteps of the father, the daughter of the mother.
Steven and I had many discussions about this, some of them tearful. Remember what I said about him being practical and having dedicated his faith to Biblical Christianity a while back? Well, this is where it came in.
The conversation I remember most vividly took place at my college. Steven was there with me for the school day, and we spent my free hours on a couch debating the issue, watching people go by.
Using the topical guide in the LDS scriptures, I had found a Bible verse where Jesus said "ye are gods." (I later found out that the way the LDS church portrays this scripture is way out of context.) Using that and the logic of growing up to become a parent like your parents, I debated with him endlessly.
Steven simplified it into something you'd see as a kid on Saturday morning. "Just imagine this," he said. "We're going to make this into a cartoon, okay? Imagine God in a lab, and he's got a petri dish that has our world in it, and he watches us through a magnifying glass. Then you die, and poof, you pop out of the petri dish. He goes, 'here you go!' and hands you a lab coat and a petri dish and tells you to make your own world. Did I get that about right?"
"In a simplified and ridiculous way, yes," I said in exasperation.
"Well, it is ridiculous!" he exclaimed, both amused and equally exasperated.
"No, its not. It makes sense," I returned to my argument. And off we went in circles about lab coats and divine inheritance.
When I left the church, I of course had to question this belief. A false prophet having taught it, I wasn't too optimistic that it would hold up.
Sure enough, upon a closer look, the Mormon "solution" to a non-human God and the question of where we come from and where we're going created more conundrums than it solved, and it completed contradicts the Bible. The Word is very clear about there being no other gods, besides ours. Not that ours is just "the only one we have anything to do with." He is the only true God.
"You are my witnesses," declares the LORD, "...Before me no god was formed, nor will there be one after me" (Isaiah 43:10).
"This is what the LORD says -- ...I am the first and I am the last; apart from me there is no God...Is there any God besides me? No, there is no other Rock; I know not one" (Isaiah 44:6,8).
"I am the LORD, and there is no other; apart from me there is no God" (Isaiah 45:5).