  This article tackles the problem of existence and classification of maximal
growth distributions on smooth manifolds. We show that maximal growth
distributions of rank$>2$ abide by a full $h$-principle in all dimensions. We
make use of M. Gromov's higher order convex integration and, on the way, we
establish a new criterion for checking ampleness of a differential relation.
  As a consequence we answer in the positive, for $k>2$, the long-standing open
question posed by M. Kazarian and B. Shapiro more than 25 years ago in [14] of
whether any parallelizable manifold admits a $k$-rank distribution of maximal
growth. We also answer several related open questions.
  For completeness we show that the differential relation of maximal growth for
rank-$2$ distributions is not ample in any ambient dimension. Non-ampleness of
the Engel and the $(2,3,5)$-conditions follow as particular cases.
