  In context of the Wolfram Physics Project, a certain class of abstract
rewrite systems known as "multiway systems" have played an important role in
discrete models of spacetime and quantum mechanics. However, as abstract
mathematical entities, these rewrite systems are interesting in their own
right. This paper undertakes the effort to establish computational properties
of multiway systems. Specifically, we investigate growth rates and growth
classes of string-based multiway systems. After introducing the concepts of
"growth functions", "growth rates" and "growth classes" to quantify a system's
state-space growth over "time" (successive steps of evolution) on different
levels of precision, we use them to show that multiway systems can, in a
specific sense, grow slower than all computable functions while never exceeding
the growth rate of exponential functions. In addition, we start developing a
classification scheme for multiway systems based on their growth class.
Furthermore, we find that multiway growth functions are not trivially regular
but instead "computationally diverse", meaning that they are capable of
computing or approximating various commonly encountered mathematical functions.
We discuss several implications of these properties as well as their physical
relevance. Apart from that, we present and exemplify methods for explicitly
constructing multiway systems to yield desired growth functions.
