  We construct a sequence of rank 3 distributions on $n$-dimensional manifolds
for any $n\geq 7$ such that the dimension of their symmetry group grows
exponentially in $n$ (more precisely it is equal to
$\operatorname{Fib}_{n-1}+n+2$, where $\operatorname{Fib}_n$ is the $n$-th
Fibonacci number, starting with $\operatorname{Fib}_1=\operatorname{Fib}_2=1$)
and such that the maximal order of weighted jet needed to determine these
symmetries grows quadratically in $n$. These examples are in sharp contrast
with the parabolic geometries where the dimension of a symmetry group grows
polynomially with respect to the dimension of the ambient manifold and the
corresponding maximal order of weighted jet space is equal to the degree of
nonholonomy of the underlying distribution plus $1$. Our models are closely
related to the geometry of certain curves of symplectic flags and of the
rational normal curves.
