It turns out, I made an error in Cracking Mathematics. Not (in this case) a mathematical or historical error, although there are plenty of those1 but an error of etiquette: my potted biography of John Horton Conway emphasised the Game of Life above the rest of his work; I imagine the book made a satisfying thud as he threw it aside.
Siobhan Roberts's biography of Conway, Genius at Play (about £13 at the time of writing), goes into rather more detail about Conway's achievements: the ATLAS of groups, monstrous moonshine and surreal numbers, to pick a few off the top of my head - and his character: magnetic, brilliant, childlike, often jerkish. He's a compelling chap, all right.
I've seen the book described as a metabiography, almost as much about Roberts's struggles to put it together as about Conway himself; that strikes me as a little unfair, but there's certainly something a little recursive about it - Conway, in his own font, inserts himself irrepressibly into the development of his own biography.
I love that it treats the maths with some attention, giving enough background to surreal numbers and the Game of Life that the reader has something to chew on and take further if so desired - although I confess I was left a bit flummoxed about the importance of the monster group. There's a limit to how much can be reasonably explained to a non-specialist reader, even in such a large book.
I see many of my own character flaws in Conway - the need to show off with number tricks, scatterbrainedness, an unwillingness to write up results - although thankfully I don't take them to such extremes. (The same, of course, goes for being good at maths.)
Genius at Play is an excellent read, and a great insight into one of the key characters in 20th century mathematics.
* Many thanks to @teakayb for the gift of the book.