The Shannon capacity of a graph is defined as $$\Theta(G) = \sup_k \sqrt[k]{\alpha(G^k)}.$$

So, $\alpha(G) \leq \Theta(G)$ but $\Theta(G)$ can be strictly greater than $\alpha(G)$. I am wondering if there is any upper bound based on the independence number itself? Specifically, are there graphs where $\Theta(G) \geq \alpha(G) + 1$? It seems like the structure of the strong product limits how much the independence number can grow. The independence number would have to grow pretty quick just for $\sqrt[k]{\alpha(G^k)}$ to get up to $\alpha(G) + 1$ for some $k$.