"Infinity Plus One, and Other Surreal Numbers," by Polly Shulman. Discover, December 1995, pages 96-105.
This article provides a fanciful, readable description of the surreal numbers, John H. Conway's extension of ordinal numbers. The story begins with a discussion of how number systems have expanded over the centuries to include not just natural numbers, but rational, irrational, complex numbers, and ordinal numbers. The construction of the surreal numbers is presented as a natural extension of this process. In addition to explaining the mathematical ideas, the article draws intriguing portraits of Conway and his fellow surreal number devotee, Martin D. Kruskal. There are no apologies for the fact that surreal numbers have no practical applications; they are appreciated on their own terms as a fascinating bit of mathematics.