Before revision:  We present a construction that shows that for an arbitrary constant 0<c<2 and sufficiently large n there exist n-dimensional Boolean networks with this property in which both the indegree and outdegree of each for each variable is bounded by two  and which nevertheless contain periodic orbits of length at least c^n.
After revision:  We present a construction that shows that for an arbitrary constant 0<c<2 and sufficiently large n there exist n-dimensional Boolean networks with this property in which both the indegree and outdegree of each variable is bounded by two and which nevertheless contain periodic orbits of length at least c^n.