Let $L$ be the length of the string. If $x$ is the length of the shortest piece, then since each of the other pieces is twice the length of the next smaller piece, then the lengths of the remaining pieces are $2x, 4x$, and $8x$. Since these four pieces make up the full length of the string, then $x+2x+4x+8x=L$ or $15x=L$ and so $x=\frac{1}{15}L$. Thus, the longest piece has length $8x=\frac{8}{15}L$, which is $\frac{8}{15}$ of the length of the string.