We prove the existence of (one-way) communication tasks with a subconstant versus superconstant asymptotic gap, which we call "doubly infinite," between their quantum information and communication complexities. We do so by studying the exclusion game [C. Perry et al., Phys. Rev. Lett. 115, 030504 (2015)] for which there exist instances where the quantum information complexity tends to zero as the size of the input n increases. By showing that the quantum communication complexity of these games

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... cales at least logarithmically in n, we obtain our result. We further show that the established lower bounds and gaps still hold even if we allow a small probability of error. However in this case, the n-qubit quantum message of the zero-error strategy can be compressed polynomially.