Let $H^\infty$ denote the Banach space of all bounded analytic functions on the open disc $\mathbb{D}$. It is easy to see that $H^\infty$ is a dual space. However, is there a Banach sapce $Y$ such that $H^\infty$ and $Y^{**}$ are isomorphic as Banach spaces?

If you only require isomorphism in the sense of an invertible, continuous linear bijection, then the answer is yes. If you require isometric linear isomorphism, the answer is no (because the unique *isometric* predual of $H^\infty$ is $L^1/H^1_0$, and $L^1/H^1_0$ is not isomorphic to any dual Banach space).

These results can be found in Wojtaszczyk's *Banach Spaces for Analysts*: see the remarks/notes at the end of Section III.E just before the exercises.