Take $X$ to be the Moore space for the group $\mathbb Z[\frac1p]$ in dimension $n$ (realized by a telescope of $n$-spheres mapping to each other vie the times-$p$ map), and take $h^*$ to be ordinary cohomology with coefficients in $\mathbb Z$.

The universal coefficient theorem gives you a sort exact sequence
$$
0\longrightarrow \mathrm{Ext}(H_n(X,\mathbb Z),\mathbb Z) \longrightarrow H^{n+1}(X,\mathbb Z)
\longrightarrow \mathrm{Hom}(H_{n+1}(X,\mathbb Z),\mathbb Z)\longrightarrow 0.
$$
The last term is zero, and so we get

$$H^{n+1}(X,\mathbb Z)=\mathrm{Ext}(H_n(X,\mathbb Z),\mathbb Z)=\mathrm{Ext}(\mathbb Z[\tfrac1p],\mathbb Z)=\mathbb Z_p/\mathbb Z,
$$
the quotient of the $p$-adics by the integers.

That same computation can be done using the Milnor sequence.
The limit of
$$
\mathbb Z\stackrel p \leftarrow\mathbb Z\stackrel p \leftarrow\mathbb Z\stackrel p \leftarrow\ldots
$$
is zero, but the $lim^1$ of that inverse system is not zero. It's again $\mathbb Z_p/\mathbb Z$.