7. $\frac{4 \sqrt{5}}{5}$  Let $A^{\prime} G \perp D E$ at $G, A^{\prime} H \perp$ plane $D E F$ at $H$, then we have $A^{\prime} H \leqslant A^{\prime} G=\frac{A^{\prime} D \cdot A^{\prime} E}{D E}=\frac{4 \sqrt{5}}{5}$.
7. In square $ABCD$ with side length $4$, $E$ is the midpoint of $AB$, and $F$ is a moving point on side $BC$. If $\triangle ADE$ and $\triangle DCF$ are folded along $DE$ and $DF$ respectively, so that $A$ and $C$ coincide at $A'$, then the maximum distance from point $A'$ to the plane $DEF$ is $\qquad$.