As Koen S points out, the longest element of an irreducible Weyl group is treated in an earlier question (in fact, it comes up in several questions). The question asked here presupposes a standard linear realization of the Weyl group, as occurs in the structure theory of a semisimple Lie algebra over $\mathbb{C}$ for instance. In this context, the standard answer given by Max is formulated as Exercise 5 at the end of Section 13 in my 1972 Springer graduate text. However one arrives at this conclusion, it obviously depends on the classification of irreducible root systems.
On the other hand, the question makes sense for any irreducible finite Coxeter group and is approached in this spirit (via the Coxeter element) toward the end of Section 3.19 in my 1990 book on reflection groups and Coxeter groups.