The author considers two-person repeated games with lack of information on one side where the role of the informed player consists exclusively of transmitting information. He shows that, even in this case, there exist Nash equilibria that cannot be achieved with a single stage of signalling. He also provides an example where the number of revealing stages required by such an equilibrium is unbounded.
Forges, F. (1984). Note on Nash equilibria in infinitely repeated games with incomplete information. International Journal of Game Theory, 13(3), 179-187. https://doi.org/10.1007/BF01769814 (Original work published 1984)