For $d=3$ the homotopy groups can be pretty elaborate.  Consider the connect-sum of some lens spaces. The universal cover embeds in $S^3$ as the complement of a cantor set (except for a few degenerate cases where you have $\mathbb RP^3$ summands).  So the homotopy-groups are pretty complicated ($\pi_2$ is finitely generated over $\pi_1$).  You could probably make an argument that this is about the worst thing that can happen for the homotopy-groups of 3-manifolds. 

You might want to phrase your question as a question about the Postnikov towers of manifolds.  Eilenberg-Maclane spaces are rarely compact boundaryless manifolds. 

edit: I guess another spin on your question could go like this. We know the fundamental groups of compact manifolds are all possible finitely presentable groups provided $n \geq 4$. So is there a sense in which the homotopy-algebras of manifolds can be anything finitely presentable? Say, for example, $\pi_2$. As a module over the group-ring of $\pi_1$, are there any restrictions beyond being finitely generated?  I suppose you could construct a compact $6$-manifold with $\pi_2$ (almost) any finitely-presented thing over any finitely-presented $\pi_1$ pretty much the exact same way $4$-manifolds with any finitely presented $\pi_1$ are constructed. I think if $H_2(\pi_1)$ is non-trivial you might run into problems following the analogous construction, in that $\pi_2$ might strictly contain the $\pi_2$ you're trying to create.