Abstract We propose a greedy interpolation approach to compute the L∞-norm of a possibly irrational L∞-function. We approximate this function by a sequence of rational L∞-functions. For this we use interpolation employing the Loewner matrix framework. In each iteration, the L∞-norm of the rational approximation is computed using established methods. Then a new interpolation point is added where the L∞-norm of the function approximation is attained. This way, the L∞-norm of the approximation converges superlinearly to the L∞-norm of the original function. We illustrate the efficiency of the resulting algorithm for various numerical examples and compare it to state-of-the-art methods.