Transfinite duals of quasireflexive Banach spaces

The transfinite duals of a space with a neighborly basis are constructed until they become nonseparable. Let $s(X)$ be the first ordinal $\alpha$ so that ${X^\alpha }$ is nonseparable. It is shown that if $X$ is nonreflexive, $s(X) \leqslant {\omega ^2} + 1$ (this is best possible) and that $\{ s(X):X{\text{separable quasireflexive of order one}}\} = \{ \omega + 1,\omega + 2,2\omega + 1,2\omega + 2,{\omega ^2} + 1\}$. A quasireflexive space $X$ is constructed so that ${X^\omega }$ is isomorphic to $X \oplus {c_0}$ and no basic sequence in $X$ is equivalent to a neighborly basis. It is shown that the ${\omega ^2}$th dual of James space and James function space are isomorphic to subspaces of one another. Also, perhaps of interest on its own, a reflexive space with a subsymmetric basis is constructed whose inversion spans a nonreflexive space.