A cardinal related to compositions of Sierpinski-Zygmund functions will be considered. A combinatorial characterization of the cardinal is given and is used to answer some questions of K. Ciesielski and T. Natkaniec. Also, a cardinal number defined by S. Shelah is shown to be equal to the bounding number under certain conditions. Finally, it is shown that the bounding number of the continuum may be strictly smaller than continuum.