We present a construction which associates an infinite sequence of Kac-Moody algebras, labeled by a positive integer n, to one single Jordan algebra. For n=1, this reduces to the well known Kantor-Koecher-Tits construction. Our generalization utilizes a new relation between different generalized Jordan triple systems, together with their known connections to Jordan and Lie algebras. Applied to the Jordan algebra of Hermitian 3 x 3 matrices over the division algebras R, C, H, O, the construction gives the exceptional Lie algebras f(4), e(6), e(7), e(8) for n=2. Moreover, we obtain their infinite-dimensional extensions for n >= 3. In the case of 2 x 2 matrices, the resulting Lie algebras are of the form so(p+n, q+n) and the concomitant nonlinear realization generalizes the conformal transformations in a spacetime of signature (p,q).