  Iterated Hopf Ore extensions (IHOEs) over an algebraically closed base field
k of positive characteristic p are studied. We show that every IHOE over k
satisfies a polynomial identity, with PI-degree a power of p, and that it is a
filtered deformation of a commutative polynomial ring. We classify all 2-step
IHOEs over k, thus generalising the classification of 2-dimensional connected
unipotent algebraic groups over k. Further properties of 2-step IHOEs are
described: for example their simple modules are classified, and every 2-step
IHOE is shown to possess a large Hopf center and hence an analog of the
restricted enveloping algebra of a Lie k-algebra. As one of a number of
questions listed, we propose that such a restricted Hopf algebra may exist for
every IHOE over k.
