It’s a common mistake to confound infinite with unbounded. A process can continue without end and still be “bounded in a nutshell”. So a sign process can pass from sign to interpretant sign to next interpretant sign ad infinitum without ever leaving a finite set of signs.
The number of questions I got about that statement tells me I should have delineated the context in which it was set a little more fully.
A sign process in this context is simply a sequence of signs, of the sort we might observe in communicational, computational, or experimental settings. For people who remember the more ancient arts of AI, cognitive science, and cybernetics, it may help to recall the orders of considerations arising in protocol analysis.
It goes with this territory to assume the formal equivalent of categorical perception. This means we can set aside the subtleties of token haecceity — the nominal distinctiveness of every individual sign instance — along with the possibility of signs being sampled from a continuous medium.
In this setting we are left with two interpretations for infinite and bounded, depending on whether the sign domain has a quantitative measure defined on it, or not. In the first case, bounded means the sequence never exceeds a finite bound in the relevant measure. In the second case, bounded means the sequence never leaves a finite set.