Can an RSA public key be generated without ever knowing the factors?
Unless you relax the restriction to allow for multi-party key generation where, if all the generating parties collude, the key can be recovered, no.

You can create something that operates like an RSA modulus. This was first proposed for use in cryptographic accumulators by Sander in "Efficient Accumulators without Trapdoor Extended Abstract".  The idea is to create a number that has at least two large prime factors that are unknown to anyone. It's known as an RSA unfactorable object (RSA UFO) and as far as I know, it's security for use in encryption or signing hasn't been thoroughly examined.

Those issues aside, the problem is the solution is by no means practical. For something with the security of a 1024 bit RSA modulus/key, you would need a 40,000 bit RSA UFO.