By AM-GM,
\(\frac{w+2x+3y+4z}{4} \ge \sqrt[4]{24wxyz}\\ 2\ge\sqrt[4]{24wxyz}\\ 16\ge24wxyz\\ \frac{2}{3}\ge wxyz\)
so the maximum value is \(\boxed{\frac{2}{3}}\)
for completeness, equality occurs when the terms are equal: \(w=2, x=1, y=\frac{2}{3}, z=\frac{1}{2}\)
