Is there an example of two reactions that catalyze one another? More than two is also helpful, but I guess it will be hard to find one.
By "catalyze one another", I mean that one of the products of the first reaction acts as a catalyst for the second reaction.
An early synthesis of borazon from hexagonal boron nitride would be one possible example. When other approaches did not work, Robert H. Wentorf Jr. added magnesium and voila! The magnesium had reacted with some of the hexagonal nitride to form $\ce{Mg3N2}$, and the latter then catalyzed the hexagonal to cubic conversion. (Today other methods have taken over.)
