Since you mentioned Kingman's subadditive ergodic theorem, you may find interesting the following *semi-uniform subadditive ergodic theorem*:

Let $T \colon X \to X$ be a continuous map of a compact metric space $X$.
If $f_n \colon X \to [-\infty,+\infty)$ is a subadditive sequence ($f_{n+m} \le f_n + f_m \circ T^n$) of upper semicontinuous functions then:
$$
\sup_{\mu} \lim_{n \to \infty} \frac{1}{n} \int_X f_n d\mu = 
\lim_{n \to \infty} \frac{1}{n} \sup_{x \in X} f_n(x)  ,
$$
where the first $\sup$ is taken over all $T$-invariant probability measures.

References:

 - Schreiber. J. Diff. Eq. 148 (1998), 334--350. 
 - Sturman, Stark. Nonlinearity 13 (2000), 113--143.
 - Morris. Proc. London Math. Soc. (3) 107 (2013) 121–150. See Appendix A.