RIP John Conway. I asked him a lot of questions when I was a postdoc at Princeton and he always had a lengthy, deep, informative answer

which had nothing to do with my question

but I got real value out of every single one.

I remember a great lecture of his filled with his usual nonstandard fanciful notation for everything and then at one point he drew a cube and said, we call this a 'cube'

He seemed totally unprepared for but also pleased by the large wave of laughter that followed

The piece of math that for me most captures Conway's wonderfully individual style is his work on "audioactive decay" -- paper behind paywall but

here's someone else writing about it http://www.se16.info/mhi/Part1.htm 

What makes it so Conway: a "normal" mathematician would see the question and say "There cannot possibly be any interesting mathematics here."

Conway, as he always did, said "Why not?"