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 $\mathbb C \mathbb P^\infty$.
the affine Grassmannian $\Omega G$ is a very nice space: it's a complex (ind-)variety, and it is stratified by finite dimensional cells. In particular, it has a natural CW-decomposition. Each cells of $\Omega G$ is isomorphic to $\mathbb C^n$, and is in particular of even (real) dimension.
Moreover, $\Omega G$ is a coadjoint orbit of the infinite dimensional Lie group $S^1\ltimes \widetilde {LG}$. Here, the tilde refers to the universal central extension of the loop group $LG$, and $S^1$ acts by reparametrizing the loops.
The inclusion $\Omega G\to Lie(S^1\ltimes \widetilde {LG} )^* $ can be composed with the projection $Lie(S^1\ltimes \widetilde {LG})^* \twoheadrightarrow (\mathfrak t_{S^1\ltimes \widetilde {LG}})^* \cong \mathfrak t^* \oplus \mathbb R \oplus \mathbb R$. It turns out that the composite lands in a translated copy of $\mathfrak t^* \oplus \mathbb R$, and so one gets map $$ \mu:\Omega G \to \mathfrak t^* \oplus \mathbb R $$ called the moment map (for the $T_{S^1\ltimes LG}$ action).
What is important, is that the space $t^* \oplus \mathbb R$ has a natural basis that is indexed by the vertices of the extended Dynkin diagram: those are the simple coroots. I will denote each cell by the moment map image in $\mathfrak t^* \oplus \mathbb R$ of its center point (in the basis of simple coroots).
Here we go:
0-dimensional cell:
$\qquad\begin{matrix}
0 - 0 - \stackrel{\stackrel{\displaystyle 0}|}{0} - 0 - 0 - 0 - 0 - 0
\end{matrix}$
2-dimensional cell:
$\qquad\begin{matrix}
0 - 0 - \stackrel{\stackrel{\displaystyle 0}|}{0} - 0 - 0 - 0 - 0 - 1
\end{matrix}$
4-dimensional cell:
$\qquad\begin{matrix}
0 - 0 - \stackrel{\stackrel{\displaystyle 0}|}{0} - 0 - 0 - 0 - 1 - 1
\end{matrix}$
6-dimensional cell:
$\qquad\begin{matrix}
0 - 0 - \stackrel{\stackrel{\displaystyle 0}|}{0} - 0 - 0 - 1 - 1 - 1
\end{matrix}$
8-dimensional cell:
$\qquad\begin{matrix}
0 - 0 - \stackrel{\stackrel{\displaystyle 0}|}{0} - 0 - 1 - 1 - 1 - 1
\end{matrix}$
10-dimensional cell:
$\qquad\begin{matrix}
0 - 0 - \stackrel{\stackrel{\displaystyle 0}|}{0} - 1 - 1 - 1 - 1 - 1
\end{matrix}$
12-dimensional cell:
$\qquad\begin{matrix}
0 - 0 - \stackrel{\stackrel{\displaystyle 0}|}{1} - 1 - 1 - 1 - 1 - 1
\end{matrix}$
14-dimensional cell:
$\qquad\begin{matrix}
0 - 0 - \stackrel{\stackrel{\displaystyle 1}|}{1} - 1 - 1 - 1 - 1 - 1
\end{matrix}$
other 14-dimensional cell:
$\qquad\begin{matrix}
0 - 1 - \stackrel{\stackrel{\displaystyle 0}|}{1} - 1 - 1 - 1 - 1 - 1
\end{matrix}$
As you can see, $ H^* (\Omega G) = H^* (\mathbb C \mathbb P^\infty ) $ for $*\le 13$. Even better: the varieties $\Omega G$ and $\mathbb C \mathbb P^\infty$ are isomorphic in complex dimensions $\le 6$. In particular, the CW-complexes $\Omega G$ and $\mathbb C \mathbb P^\infty$ are isomorhpic in dimensions $\le 13$. Taking classifying spaces, we get that the CW-complexes $G$ and $K(\mathbb Z,3)$ are isomorphic in dimensions $\le 14$.