  Suppose $((\cdots((x^{2}-c_{1})^{2}-c_{2})^{2}\cdots)^{2}-c_{k-1})^{2}-c_{k}$
splits into linear factors over $\mathbb{Z}$ and $c_{k}\neq0$. We show that for
each $j$ and each prime $p$, if $p\leq2^{j-1}$ then $p$ divides $c_{j}$.
Consequently, $$\ln c_{j}>\frac{1}{4}\cdot2^{j}\,\,\mathrm{for}\,j\geq5$$ If we
also have $p\equiv3\,(\mathrm{mod\,4)}$ then $p^{2^{j-\left\lceil \lg
p\right\rceil }}$ divides $c_{j}$. Consequently, if $k\geq3$, there exists some
absolute constant $\lambda>0$ so that, $$\ln c_{j}>\lambda
k2^{j}\mathrm{\,\,for\,all\,}j$$ These estimates argue against the possibility
of explicitly constructing polynomials of the given form for large $k$, as the
coefficients quickly become too large to manipulate.
