$\epsilon$ - Nash Equilibrium exceeds Nash Equilibrium
It is quite simple: if you enlarge the set of possibilities you can get better values.

Here, in place of requiring the possibilities to be equilibrium we require them to be almost equilibrium, a larger set of possibilities, therefore a higher possible value.

Here is a simple example:

$$\left[\begin{array}{cc}
5/5 &amp; 0/6 \\
6/0 &amp; 1/1 \\
\end{array}\right]$$

Let $\epsilon = 2$. Now $5/5$ on the top left corner is an $\epsilon$-equilibrium where as the equilibrium is $1/1$.