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.