(Fellow bloggers Scott Aaronson and Terry Tao have already posted about John Conway,
here and here. I suspect there will be others and when they do I will add it here.
ADDED LATER: nice xkcd here
John Conway is a great example of how the line between recreational math and serious math is .... non-existent? not important? Take our pick.
1) Conway invented Surreal Numbers. These can be used to express infinitely big and infinitely small numbers. One can even make sense of things like square root of infinity. Conway's book is called On Numbers and Games (see here and here) Two free sources: here and here.
Note that Conway defined surreals in terms of games. Are they fun games? Probably not, but they are games!
2) Conway's Game of Life (you really do need to use his name, note the contrast between The game of life here and Conway's Game of Life here
The game is simple (and this one IS fun). You begin with some set of dots placed at lattice points, and a set of rules to tell how they live, die, or reproduce. The rules are always the same. Different initial patterns form all kinds of patterns. Sounds fun! Is it easy to tell, given pattern P1 and P2 whether, starting with P1 you can get to P2. No. Its undecidable.
So this simple fun game leads to very complicated patterns.
And nice to have an undecidable problem that does not mention Turing Machines. (I will tell the students it is undecidable this semester, though I won't be proving it.)
This is the ultimate book on NIM games.
4) The above is probably what the readers of this blog are familiar with; however, according to his Wikipedia page (see here) he worked in Combinatorial Game Theory, Geometry, Geometric Topology, Group Theory, Number Theory, Algebra, Analysis, Algorithmics and Theoretical Physics.
He will be missed.