It is not quite clear what you mean by calculate. This limit is a familiar, classical object: it is called the Green function of the complement of the Julia set, or
the equilibrium potential of the Julia set. It is positive and harmonic in the complement of the filled in Julia set of $P$, has asymptotics $\log|z|$ at infinity,
and equals zero on the filled in Julia set (for all $b$ except possibly one).
These properties define it uniquely. This limit is also subharmonic in the plane,
and its Riesz measure is the measure of maximal entropy corresponding to $P$.
It can be also described in terms of solution of the Bottcher equation at infinity. This function is one of the main tools of Holomorphic dynamics of
For an introduction to the study of this object, see
Invariant sets under iteration of rational functions.
Ark. Mat. 6 1965 103–144 (1965).
MR0762431 Douady, A.; Hubbard, J. H. Étude dynamique des polynômes complexes. Partie I. Publications Mathématiques d'Orsay [Mathematical Publications of Orsay], 84-2. Université de Paris-Sud, Département de Mathématiques, Orsay, 1984. 75 pp.
There are exactly two cases when this limit is an elementary function: when $P$
is a monomial and when $P$ is a Chebyshev polynomial. In all other cases one
has only some infinite expression (a series, an infinite product etc.)