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Counting processes with multiple randomness, as defined in this article, differ essentially from known stochastic processes, such as doubly stochastic Poisson processes and non-homogeneous Poisson processes. By introducing general results concerning multiple randomness in a random experiment performed to count the number of ``events'' and providing specific examples to illustrate properties of the counting processes with multiple randomness, this study aims at demonstrating the existence of phenomena modeled by such processes. The specific examples are popular models taken from queuing theory, which may help the reader to understand the general results. Modeled by other stochastic processes, phenomena with multiple randomness similar to queuing phenomena modeled by counting processes with two-fold randomness may appear in various experiments concerning practical applications of sciences and engineering disciplines but may not have been identified. It may be reasonable to expect more phenomena with multiple randomness to be identified, which may help scientists and engineers to explain weird phenomena and solve puzzling problems in the real world.