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Exactly solvable rationally-extended radial oscillator potentials, whosewavefunctions can be expressed in terms of Laguerre-type exceptional orthogonalpolynomials, are constructed in the framework of $k$th-order supersymmetricquantum mechanics, with special emphasis on $k=2$. It is shown that for$\mu=1$, 2, and 3, there exist exactly $\mu$ distinct potentials of $\mu$thtype and associated families of exceptional orthogonal polynomials, where $\mu$denotes the degree of the polynomial $g_{\mu}$ arising in the denominator ofthe potentials.

Higher-order SUSY, exactly solvable potentials, and exceptional orthogonal polynomials