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Multiple randomness is a random phenomenon appeared in experiments concerning practical applications of sciences and engineering disciples. But phenomena with multiple randomness are ignored in the existing literature, because correctly derived results in mathematics are used to interpret, incorrectly, outcomes obtained by experiments designed to solve practical problems in the real world, leading to questionable theorems in practice. Multiple randomness is characterized by unique properties of the sample space consisting of all possible outcomes obtained by the corresponding random experiment and differs essentially from any known phenomenon observed. By introducing general results concerning multiple randomness in a random experiment performed to count the number of ``events'' and providing specific examples to illustrate the properties of multiple randomness in counting processes, this study aims to demonstrate the existence of such phenomena. The specific examples are popular models taken from queuing theory, which may help the reader to understand the general results. It is reasonable to expect more phenomena with multiple randomness to be identified in other stochastic processes and in other application fields of sciences and engineering disciples, which may help scientists and engineers to explain weired phenomena and solve puzzling problems in the real world.