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This question has been bothering me for quite a while, so let me ask it here.
Is there a first-countable compact space $X$ with uncountable Cantor-Bendixson index?
By a Cantor-Bendixson index I mean the smallest ordinal for which the process of taking the derived set stabilises.
A remark: the answer is no for ordinals with the order topology as $\omega_1+1$ is not first countable.
Here is an example. Let $X$ be $\omega_1+1$, topologized such that $U\subseteq\omega_1$ is open iff it is open in the order topology but the only open set containing $\omega_1$ is the entire space $X$. Then $X$ is compact and first-countable, and it is easy to see that for all $\alpha$ the $\alpha$th derived set of $X$ is the same as the $\alpha$th derived set of $\omega_1+1$ with the order topology, so $X$ has Cantor-Bendixson index $\omega_1+1$.
I don't know whether there exists a Hausdorff example.
