If a Banach space $X$ is not reflexive, then you have an infinite ascending chain of (continuous) dual spaces: $X’$, $X’’$, $X’’’$, etc. None of these are isomorphic to each other or to $X$.
My question is, can you have an infinite descending sequence of dual spaces? That is, is it possible for $X$ to be isomorphic to the dual space of some space $X_1$, which is isomorphic to the dual space of some space $X_2$, etc., such that none of these spaces are isomorphic to each other or to $X$? Or must the chain always stop at some space which is not isomorphic to any dual space?