If we consider crystallographic root systems, then for each $k$ such that $n \leq k \leq d-1$ where $d$ is the Coxeter number, it seems to be the case that there is exactly one root of height $k$ with nonzero coefficients with respect to each simple root. 


It seems to work if you check it case by case, but I would be interested to hear of a general way of arguing the point.



roots in a root system which have nonzero coefficients with respect to each simple root
This does not hold in general. In type E6, for example, there are roots for which the "middle" simple root has multiplicity 2 and so does one of the two other simple roots corresponding to a vertex adjacent to it, while all the other simple roots have multiplicity one. 


