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I am aware of the frequentist and bayesian interpretations of statistics. I prefer Bayesian because I think it's closer to how people think, and because we in practice often can't rerun a trial a million times to estimate the probability.
But are there any other interpretations besides those two?
I think we should distinguish between "interpretations of probability" and "interpretations of statistics" (or more adequately- "statistical argumentation"). Both, are naturally tightly related.
Regarding interpretations of probability- I like the following criteria [a]:
Any interpretation out there can be classified according to these aspects.
Regarding statistical arguments, the classics are indeed the frequentist and the Bayesian but there are more out there-- fiducial, fuzzy logic. Indeed, there are infinitely many ways to argument with data, but the two you mentioned are preferred in the sense they are considered more "convincing".
[a] [Williamson, Jon. "Philosophies of probability." Handbook of the Philosophy of Mathematics 4 (2009).]
Fisher had his own interpretation, called Fiducial probability. There is a school of Likelihood theorists, who try do something similar to Bayesian but don't use priors. They just compute and report the likelihood function. (Misguided in my view.) Apart from probability there are some approaches to uncertainty (all wrong-headed in my view) that are used in computer science. Possibility theory, Dempster-Shafer theory, rough sets, Desert-Smaradanche theory.
