Given a simple Lie group $G$, you can check how far $G$ is from beeing a $K(\mathbb Z,3)$ by looking at the place where the affine vertex gets glued onto the Dynkin diagram, and measuring the length of that tail. For $E_8$, it's the longest, and so $E_8$ is the best possible approximation to a $K(\mathbb Z,3)$.
$$\bullet - \bullet - \stackrel{\stackrel{\displaystyle\bullet}|}{\bullet} - \underbrace{\bullet - \bullet - \bullet - \bullet}_{\text{long tail}} - \circ$$
This is done by labelling the cells of the affine Grassmannian $\Omega G$ by data from the dynlin Diagram, and checking how far you need to go for $\Omega G$ to start looking different than $\mathbb C \mathbb P^\infty$.
I'll elaborate once I get some more time...