I want just to mention a framework in which we (Dikran Dikranjan, Anna Giordano Bruno and me) are going to redefine many notions of entropy.

The idea of prof. Dikranjan was to define the category of normed semigroups. Indeed, a normed semigroup is just a semigroup $S$ with a norm
$$v:S\to \mathbb R_{\geq 0}$$
such that $v(xy)\leq v(x)+v(y)$. The morphisms in the category are just semigroup homomorphisms such that the norm of the image is $\leq$ than the norm of the original point.

In this category one can define a notion of entropy of an endomorphism $\phi:(S,v)\to (S,v)$. In fact one takes
$$h(\phi)=\sup \left( \lim_{n\to\infty}\frac{v(x\phi(x)\dots\phi^{n-1}(x))}{n}: x\in S\right ) .$$
It is interesting to notice that already at this level, the above entropy function satisfies some good properties. Furthermore, it turns out that many of the usual notions of entropy (topological, algebraic, mesure-theoretic entropy, ...) for endomorphisms or automorphisms can be defined using a suitable functor from the category in which they are defined to the category of normed semigroups (the semigroup can be the set of subset with intersection (or sum in groups), the set of open covers with intersection, ... the norm can be $\log$ of the cardinality, measure, minimal cardinality of a subcover ...).

Let me conclude remarking that if you are looking for lists of axioms for entropy functions you should look to the following papers:

L. N. Stojanov, Uniqueness of topological entropy for endomorphisms on compact groups, Boll. Un. Mat. Ital. B 7 (1987) no. 3, 829–847. (axiomatic char. of topological entropy on compact groups)

D. Dikranjan and A. Giordano Bruno, Entropy on abelian groups, preprint; arXiv:1007.0533.
(axiomatic char. of algebraic entropy on discrete groups)

L. Salce, P. Vamos, S. Virili, Length functions, multiplicities and algebraic entropy, Forum Math. (2011)
(axiomatic char. of a notion of algebraic entropy on modules)

The above characterizations are discussed in the following survey,

D. Dikranjan, M. Sanchis, S. Virili, New and old facts about entropy in uniform spaces and topological groups, Topology appl. (2012) 1916-1942