I agree that it is kind of weird that he doesn't want to give you a copy. However, the reason might be -- since you say you never signed anything -- that you don't actually have a written lease and so there is nothing for him to give you. If there comes a time when he wants to enforce a provision of the lease, he certainly can't take any adverse action against you without giving you a copy of the lease he claims he's relying on. In the meantime, if you think you want to move out of this place, I would consult with an attorney about what your rights are in the event you don't have a lease (in other words, perhaps in your state that would make you a "month to month" tenant and you could get out on 30 days notice).
If there is something specific that you have a question about, I would put your question to the landlord in writing and ask him to give you a copy of the lease along with his answer. That might move things along.
You don't say whether you are having a specific problem with your tenancy (other than that you don't have a copy of your lease agreement). But if you don't remember signing a lease, then there is a chance that you didn't. But you haven't described any situation that would put you in a position to ask a court to make him give you a copy of the lease. If you are involved in a court proceeding, you can always ask the court to make him give you a copy of the lease.
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It does sound a little shady, but your landlord is required by law to give you a copy. But this sounds like something that a simple letter from an attorney might resolve. If the landlord doesn't respond, then you could probably sue to rescind the contract. It will depend upon the facts as a whole.