The problem of packing $n$ equal circles into a unit square, or equivalently, finding the largest possible minimal distance between $n$ points in a unit square, produces some high-degree polynomials (the minimal polynomials of the radii of circles). For example, for $n=11$ the minimal polynomial has degree 18, and for $n=13$ it has degree 40 (cf. http://www.inf.u-szeged.hu/~pszabo/Pub/45survey.pdf, page 17).