With the particular rule shown, the behavior always eventually stabilizes—though sometimes only after an astronomically long time.
But it is quite possible to find symbolic systems where this does not happen, as illustrated in the pictures below. Sometimes the behavior that is generated in such systems has a simple repetitive or nested form. But often—just as in so many other kinds of systems—the behavior is instead complex and seemingly quite random.
The behavior of various symbolic systems starting from the initial condition e[e[e][e]][e][e]. The plots at the bottom show the difference in size of the expressions obtained on successive steps.