The study of dynamical systems suggests that an important indicator for their classification is the quantity of information that is needed to describe their orbits (here the orbits are transated into symbolic sequences by some geometrical construction). This leads to a definition of a complexity of a single orbit. This notion is flexible enough to give a refinement of the classical definition of entropy of a system. This concept is particularly interesting for systems with zero entropy (for motivations and some results see [S. Galatolo, Complexity, initial condition sensitivity, dimension and weak chaos in dynamical systems, Nonlinearity 16 (2003) 1219–1238]).