= Three-stage accumulation tower
A subset of the plane can have three successive strict changes under taking its set of <accumulation points>. For example let
$$
A=\left\{\left(\frac1n+\frac1{n^2m},\frac1{n^2m^2k}\right):n,m,k\geq1\right\}.
$$
Put $B=\{(1/n+1/(n^2m),0):n,m\geq1\}$ and $C=\{(1/n,0):n\geq1\}$. Then $A'=B\cup C\cup\{0\}$, $A''=C\cup\{0\}$ and $A^{\prime\prime\prime}=\{0\}$, where $0$ denotes the origin. For fixed $n,m$, increasing $k$ gives a point of $B$; increasing $m$ gives a point of $C$; increasing $n$ gives the origin. These are all possible limits: in any <convergent sequence> of distinct points, either $n$ is unbounded, or a subsequence fixes $n$ and has unbounded $m$, or a subsequence fixes both and has unbounded $k$. The first-coordinate cluster <intervals> $[1/n,1/n+1/n^2]$ are disjoint and separated away from the origin. Repeating this classification after each differentiation proves the displayed derived sets. Each <derivative> removes one layer of <isolated points>, and all three displayed changes are strict.
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