Invented and built in his undergraduate years, as seen in this tweet (also here):
I found some supporting links including some words about it from Conway himself:
written by Siobhan Roberts, who wrote a longer version here (pdf).
Of course one must mention MONIAC, at least, as well (there are three MONIAC machines in the world)
When it debuted back in November 1949, the leading thinkers at the London School of Economics crammed into the seminar room, some having come just to laugh, others gaping in amazement at the thing in the middle of the room, which had been cobbled together in a garage, with a pump cannibalized from an old Lancaster bomber.
Water flows through a series of clear pipes, mimicking the way that money flows through the economy. It lets you see (literally) what would happen if you lower tax rates or increase the money supply or whatever; just open a valve here or pull a lever there and the machine sloshes away, showing in real time how the water levels rise and fall in various tanks representing the growth in personal savings, tax revenue, and so on. This device was state of the art in the 1950s, but it looks hilarious now, with all its plumbing and noisy pumps.
Though its tempting to view the Phillips machine as a relic of a bygone era, in one way its just the opposite; theres something about it as fresh as the day it began gurgling. Look at its plumbing diagram. Its a network of dynamic feedback loops. In this sense the Phillips machine foreshadowed one of the most central challenges in science today: the quest to decipher and control the complex, interconnected systems that pervade our lives.
For those wishing to learn more about Professor A. W. H. Bill Phillips, his contributions to economics, and his remarkable machine, see:
Leeson, R., ed. (2000) A. W. H. Phillips: Collected Works in Contemporary Perspective. Cambridge University Press, Cambridge, UK.
(hat-tip to @Maurici_Carbo)