One example other than the transpose is Arveson's example of a non-contractive unital positive map (see Paulsen's book, Example 2.2). Since any unital completely positive map is contractive, Arveson's map cannot be completely positive. The example is like this: let $\mathcal{S}$ be the operator system
$$
\mathcal{S}=\text{span}\{1, z, \bar{z} \} \subset C(\mathbb{T}),
$$
and let $\phi:\mathcal{S}\to M_2(\mathbb{C})$ be given by
$$
\phi(a+bz+c\bar{z})=\begin{bmatrix}a&2b\\ 2c& a\end{bmatrix}.
$$
This example might not be satisfactory for some operator algebraists because it cannot be extended to an example with a C$^*$-algebra domain. A way to produce examples between C$^*$-algebras would be this: we use Choi's theorem that characterizes completely positive maps $\varphi:M_n(\mathbb{C})\to B(H)$. Choi's result asserts that $\varphi$ is completely positive if and only if the matrix
$$
\left[\varphi(e_{kj})\right]\in M_n(B(H))
$$
is positive, where $\{e_{kj}:\ k,j=1,\ldots,n\}$ are the standard matrix units. The only caveat is that one needs to produce positive maps on the matrices by showing explicit computations on the matrix entries, and this might not be that easy. I'm familiar with examples on small operator systems, where positivity of the elements can be established with simpler formulas, but I cannot see right away a path to do this explicitly for full matrix algebras.
