  A.Regev proved that the codimension growth of an associative PI-algebra is at
most exponential. The author established a scale for the codimension growth of
Lie PI-algebras, which includes a series of functions between exponential and
factorial functions. We prove that the same scale stratifies the ordinary
codimension growth of Poisson PI-algebras satisfying Lie identical relations.
As a byproduct, we obtain a new bound on the codimension growth of Lie
PI-algebras in terms of complexity functions. We also study very fast
codimension growth of some Poisson algebras without Lie identities. We
essentially use techniques of exponential generating functions and growth of
respective fast growing entire functions.
