  We investigate monogenicity and prime splitting in extensions generated by
roots of iterated quadratic polynomials. Let $f(x)\in\mathbb{Z}[x]$ be an
irreducible, monic, quadratic polynomial, and write $f^n(x)$ for the
$n^{\text{th}}$ iterate. We obtain necessary and sufficient conditions for
$f^n(x)$ to be monogenic for each $n$. We use this to construct multiple
families where $f^n(x)$ is monogenic for every $n>0$.
