I doubt this is what you seek, but the minimal polynomial for a packing of $n$
congruent disks in a square can have arbitrarily high degree:

> Szabó, Péter Gábor, Mihály Csaba Markót, and Tibor Csendes. "Global optimization in geometry—Circle packing into the square." In *Essays and Surveys in Global Optimization*, pp. 233-265. Springer, Boston, MA, 2005. [PDF download](http://www.inf.u-szeged.hu/~pszabo/Pub/45survey.pdf).

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<sup>
The minimal polynomial for $n=13$. p.17 of Szabó et al.
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The minimal polynomial is derived from a series of quadratic equations describing
the circle contacts.
Whether these polynomials are "naturally occurring" is a judgement call.

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