I doubt this is what you seek, but the minimal polynomial for a packing $n$ 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.

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          [![N13][1]][1]
          ^{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.

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.

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          [![N13][1]][1]
          ^{The minimal polynomial for $n=13$. p.17 of Szabó et al.}

---

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.