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While it is well known that feedback does not increase the capacity of discrete memoryless channels, this note examines a discrete memoryless channel with a cost constraint at the output. When that cost constraint has memory, i.e., depends on several past channel outputs, it is demonstrated that the gap between the feedback and non-feedback capacities can be unbounded. One motivation for such channel output constraints with memory comes from cognitive radio applications in which the constraints depend on how the cognitive radio activity interferes with the primary user.