Tensor products only describe the separable (i.e., unentangled) states.

As I understand it, that is only true when dealing with density matrices. For example, |0>⊗|0> + |1>⊗|1> is entangled and has tensor products. I think that Xcelerate is correct in saying that all combinations of the basis vectors form the basis of the multi-particle Hilbert space, as a single-particle wavefunction is just a vector/ket.

Sure, the tensor product space has a basis that is formed by the tensor products of all pairs of basis elements. But this is different from saying that any particular vector in the space is a tensor product of elements from the individual spaces.

But I'm possibly just misunderstanding what you're saying.

Fair enough, I should have been more precise: By "all possible tensor products" I actually meant all elements in the tensor product space, including all linear combinations of tensor products of single-particle states.