6. Let the side length of the lateral edges of a regular quadrilateral pyramid be 1. Then the maximum value of its volume is $\qquad$

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6. $\frac{4 \sqrt{3}}{27}$.

As shown in Figure 2, let the center of the base of the regular quadrilateral pyramid $P-ABCD$ be $H$, and $\angle PAH = \theta$. Then
$$
\begin{array}{l}
AH = \cos \theta, \\
PH = \sin \theta, \\
V = \frac{2}{3} \cos^2 \theta \cdot \sin \theta.
\end{array}
$$

Let $\sin \theta = x$. Then $V = \frac{2}{3}(x - x^3)$.
Setting $V' = 0$, we get $x = \frac{\sqrt{3}}{3}$.
Thus, $V_{\max} = \frac{4 \sqrt{3}}{27}$.