jmc

Since $x,$ $y,$ and $z$ are nonnegative real numbers such that $x+y+z=1,$ $0\le x,$ $y,$ $z\le 1.$ Then $y^2\le y$ and $z^3\le z,$ so $$x+y^2+z^3\le x+y+z=1.$$Equality occurs when $x=1,$ $y=0,$ and $z=0,$ so the maximum value is $\boxed{1}.$