What is the application of combinatorial game theory
I think Kaveh's comment is the correct answer: applications? We don't need no applications.

But despite all that, combinatorial game theory does appear to have some applications in error correcting codes. See Conway and Sloane, "Lexicographic codes: Error-correcting codes from game theory", IEEE Trans. Inf. Th. 1986.

More simply, if you are willing to think of nim-addition (bitwise exclusive or) as something that came out of combinatorial game theory, then there are plenty of applications, for instance in tabulation hashing, or as one of the components of Schieber and Vishkin's lowest common ancestors data structure, or as one of the components of some block cipher systems.