The original idea of the discrete Choquet integral on nonnegative reals is based on a distinguished permutation making the inputs reordering nonincreasingly or nondecreasingly. This permutation generates the weights from the given capacity, and then a weighted arithmetic mean yields the value of the corresponding Choquet integral. Since the permutation relates to a maximal chain, we introduce a concept of MCC-integrals using the setting of maximal chains on the power set of a finite set. This approach provides a unified framework for several integrals of nonnegative vectors with respect to games. We study basic properties and various representations of these MCC-integrals subsuming known results for the Choquet integral as a special case. For supermodular games it is shown that the Choquet integral is the minimum of all product-based MCC-integrals. Further, we discuss symmetric and asymmetric extensions for real-valued inputs providing new insight into the construction of fuzzy integral quadruplets. During the way, we point out certain inaccuracies in the literature.