There is a partial order on a root system given by $\alpha \prec \beta$ if and only if $\beta-\alpha \in R^{+}$, and Humphreys (10.4. Lemma A) proves that there is a unique maximal root with respect to this ordering.
There is another notion of maximality of a root in terms of its length: Since the reflection $\sigma_{\alpha}$ can be factored into a product of simple reflections, the length of its reduced factorization into simple roots is an invariant (and it can be determined by counting the number of positive roots that are sent to negative roots under $\sigma_{\alpha}$). So the "maximal" root under this notion is the root $\alpha \in R$ such that $\sigma_{\alpha}$ satisfying $\sigma_{\alpha}(R^{+}) = R^{-}$. I don't even know if the root that is maximal in this sense is unique.
Are these two notions related?