Let $p$ be a prime and $n \in \mathbb N$. Does there exist a closed manifold of type $n$ (i.e. a closed manifold $X$ such that $\widetilde{K(n)}_\ast X \neq 0$ but $\widetilde{K(m)}_\ast X = 0$ for $m < n$, where $K(n)$ is the $n$th Morava $K$-theory at the prime $p$)?

When $n= 0$ the answer is yes. When $p = 2$ and $n = 1$ we can take $\mathbb R \mathbb P ^2$.

Other than that, I'm not sure.

It is worth noting that when $n \geq 1$, a closed manifold of type $n$ can't be orientable (since if it's orientable, then by Poincare duality its reduced rational homology is nonvanishing).