To construct J. H. Conway's look-and-say sequence, begin by putting down a 1 as the first entry. The other entries are found by saying the previous entry aloud, and writing what you hear.
1 11 21 1211 111221 312211 (previous entry was three 1's, two 2's and one 1) ...
Conway provides his usual fantastical analysis in The Weird and Wonderful Chemistry of Audioactive Decay [Eureka 46, 5-18], where he demonstrates several otherworldly properties of this sequence. One was this: the ratio of the lengths of consecutive entries has a limit, $\lambda$. Furthermore, $\lambda$ is the root of a polynomial of degree 71.
Now, when I was in high school we were taught the quadratic formula and told there is a cubic formula, but you don't have to learn it. Why? "You won't be needing it." And mostly I've found that to be true. Am I wrong, or do high-degree polynomials rarely occur (in uncontrived settings)?
What are some other examples of useful roots of polynomials of high degree? Power series and the like can obviously produce useful polynomials of arbitrarily large degree, but I'm looking for surprises such as the degree-71 polynomial at the heart of the look-and-say sequence above.