Biological signaling networks can exhibit rich response dynamics including ultrasensitivity, adaptation to persistent stimuli and oscillations. Previous modeling efforts have considered the proteins in these networks as two-state entities and their total levels as fixed quantities. However, inside the cell, most molecules are in constant flux because of various processes such as degradation, synthesis, binding of scaffold proteins and release from vesicles. The resulting freedom in the amount of signaling protein that is available for signaling has not been explored. Here, we analyze the response dynamics of a signaling protein when it enters the signaling pool in one state (modified or unmodified) and exits in both states. When the exit rates of these two states are comparable, a persistent stimulus results in step responses and can produce ultrasensitivity, as shown previously. However, we find that when the exit rates are imbalanced, the signaling protein gives transient responses to persistent stimuli even though the system does not have any explicit feedback. Further, these rates determine the signal range over which the system is responsive. Building small networks from signaling proteins with different exit rates, we show that these systems can exhibit rich behavior. Taken together, these findings indicate that altering the total level of signaling proteins can significantly change their response and provide additional richness in system dynamics. We discuss relevant biological examples in which regulating total protein levels could be exploited to alter signaling behavior.