  We exhibit the first examples of contact structures on $S^{2n-1}$ with $n\geq
4$ and on $S^3\times S^2$, all equipped with their standard smooth structures,
for which every Reeb flow has positive topological entropy. As a new technical
tool for the study of the volume growth of Reeb flows we introduce the notion
of algebraic growth of wrapped Floer homology. Its power stems from its
stability under several geometric operations on Liouville domains.
