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f(x,n) = f(x,n-1) ^ x

f(x,1) = x

c_n is such that f(c_n,n) = 2

c = lim c_n

Do you mean f(x,n-1)^x or x^f(x,n-1)? With the first definition, c=1, the second, c=sqrt(2). I'm still not seeing the connection to omega.