Taking only the basic decision theory (theories?) based on maximization of expected utility, the answer depends on
(i) the definition of what is considered Bayesian and
(ii) whether a theory that has both a Bayesian and a nonbayesian interpretation or a nonbayesian theory that can be approximated by a Bayesian one count as counterexamples.
Chapter 10 of Martin Peterson's "An Introduction to Decision Theory" (Cambridge, 2009) from Tim's answer characterizes a Bayesian decision theory as follows:
- Subjective degrees of belief can be represented by a probability function
defined in terms of the decision maker's preferences over uncertain
- Degrees of desire can be represented by a utility function defined in the
same way, that is, in terms of preferences over uncertain prospects.
- Rational decision makers act as if they maximise subjective expected
Regarding (i): The step that seems particularly/genuinely Bayesian* to me is 1. but not necessarily 2. or 3. However, one could say all of them are as Bayesian as it gets because of the persons who developed and advanced such theories (Ramsey, de Finetti, Savage).
Regarding (ii): The framework above might be broad enough to cover theories of maximization of expected utility based on probability distributions arrived at without the use of Bayesian thinking. (I think this is commonly seen in microeconomics textbooks, at least on the introductory level; they often take probability distributions as given, leaving it up to the user to figure out how to come up with them in reality.) E.g. in step 1. one could develop a model of uncertain prospects and use frequentist or fiducial techniques to estimate its parameters without ever invoking Bayesian arguments. One could then proceed with steps 2. and 3. as above. Now, this could most likely be approximated by the framework above, and in that sense this would (almost?) count as a Bayesian decision theory. On the other hand, does that necessarily invalidate this as a counterexample?
(Work in progress)
*The definition and delimitation of the term genuinely Bayesian is certainly debatable and I may be making a mistake here.