To construct J. H. Conway's [look-and-say sequence][1], 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)?

To me, anything over degree 6 seems exotic. What are some other examples of useful roots of polynomials of high degree? [1]: http://en.wikipedia.org/wiki/Look-and-say_sequence