Examples for directed graphs with super polynomial cover time
Yes, it's not too hard to construct such a graph.  Consider a path $v_1 \to v_2 \to v_3 \to \dots \to v_n$ containing $n$ vertices.  Now add a new "dead-end vertex" $w$, and add edges $v_1 \to w$, $v_2 \to w$, $v_3 \to w$, etc.

If you do a random walk starting at $v_1$, the probability that you reach $v_n$ is $1/2^{n-1}$, as you have to get lucky and move in the right direction at each of the vertices $v_1,\dots,v_{n-1}$ to avoid getting stuck at the dead end.  Therefore, the cover time will be $\Omega(2^n)$.



Notice how things would change if we converted this into an undirected graph.  In an undirected graph $w$ would no longer be a dead end: if we hit $w$, we're not stuck (we still have a chance to get back to where we want to be, by backtracking our steps).  In the directed graph, $w$ is a dead end: once you hit $w$, you're done and you have no hope of making it to where you want to be.