Multithreaded parameterized programs with superexponential shared memory size in the number of threads?

In this question, a program means a parameterized multithreaded program with the interleaving semantics, a finite number of per-thread states (which may depend on the number of threads), and finite number of shared states (which may depend on the number of threads).

A shared state is a valuation of the shared memory, and a per-thread (in other terminology, local) state is the valuation of thread-local memory (we assume no stack). Interleaving semantics means that the actions of the threads are interleaved on a single processor and a thread has rw-access to the shared memory and its own local memory and no access to the local memories of the other threads. Parameterized means that we conside a family of programs generated from a finite-desciption template such that the $ n$ th member of the family has $ n$ threads (which typically coincide up to the thread identifier).

To the best of my knowledge, for such a program, the size of the shared state-space is anywhere between constant (e.g., for a single boolean lock variable) and exponential (e.g., Peterson mutual exclusion protocol) in the number of the threads $ n$ .

Is there any well-known academic program in which the size of the shared state-space grows superexponentially in $ n$ ?