  It is shown that for every $k\in\mathbb{N}$ and every spreading sequence
$\{e_n\}_{n\in\mathbb{N}}$ that generates a uniformly convex Banach space $E$,
there exists a uniformly convex Banach space $X_{k+1}$ admitting
$\{e_n\}_{n\in\mathbb{N}}$ as a $k+1$-iterated spreading model, but not as a
$k$-iterated one.
