Many times, when I'm busy researching a PE problem, I'll end up finding something that Conway worked on. The man had his fingers in everything, it seems, to do with recreational mathematics. I wonder how many problems here were directly inspired by his work, maybe pulling ideas from the book he did with Berlekamp and Guy, Winning Ways For Your Mathematical Plays. I'd start listing some of them here, but that would constitute a spoiler. Better to end up making the connection yourself, I think.
As for many others, one of my earliest computing activities was writing a program for the Game of Life. In my case, that was the mid-70s, before I had access to a microcomputer with a video screen, so my program spit out progressive generations on 132 column green fan-fold paper on a mini-computer's line printer. So much paper
But it was only when I started working through Project Euler, especially when I started solving the impartial games problems, that I realized just how many fun ideas Conway had, and what an inventive mind he had.
RIP, Professor.
