TITLE: Conway's lesser-known results
QUESTION [230 upvotes]: John Horton Conway is known for many achievements:
Life, the three sporadic groups in the "Conway constellation," surreal numbers, his "Look-and-Say" sequence analysis, the Conway-Schneeberger $15$-theorem, the Free-Will theorem—the list goes on and on.
But he was so prolific that I bet he established many less-celebrated
results not so widely known. Here is one: 
a surprising closed billiard-ball trajectory in a regular tetrahedron:

          


          

Image from Izidor Hafner.



Q. What are other of Conway's lesser-known results?


Edit: Professor Conway passed away April 11, 2020 from complications of covid-19:
https://www.princeton.edu/news/2020/04/14/mathematician-john-horton-conway-magical-genius-known-inventing-game-life-dies-age

REPLY [3 votes]: Conway polynomials are irreducible polynomials that provide a basis for finite fields of order $p^n$.
While there is a unique finite field of order $p^n$, there are many ways of representing its elements as polynomials, all resulting in the same field (up to isomorphism).
Conway polynomials provide a standardize choice of basis.
They are nice because they satisfy a certain compatibility condition with respect to subfields (i.e., fields of order $p^m$ with $m$ dividing $n$).
Formally, the Conway polynomial for $\mathbb{F}_{p^n}$ is defined as the lexicographically minimal monic primitive polynomial of degree $n$ over $\mathbb{F}_p$ that is compatible with the Conway polynomials of all its subfields (see Wikipedia for more details).
Of course, imposing the lexicographical ordering is a convention and is necessary to make them unique.
Conway polynomials are very useful when performing computations using computer algebra systems.
They also provide portability among different systems.
Computing them in general is hard, however for many small cases they have been tabulated
(e.g., see the extensive tables computed by Frank Lübeck).
These tables are available, for example, in Sage.