My two most common:

• My roommate and I use the intuitions behind Bayesian Search Theory explicitly when we are looking for something in the house or notice that the other is, though we do not use the math. Briefly: if you've already searched the kitchen thoroughly, it's probably somewhere else (or under something absurd) even though you remember putting it down in the kitchen.
• I use the intuitions in hidden Bayes-structure at work quite a lot as a framework for thinking about our projects. Supposedly this product is improving users' lives - it's essentially making decisions about what to show or not show the user, what to do or not do - there's some average behavior that we decide it shouldn't always do - BAM find the evidence that the product we ship uses to update its priors and act differently than average, then ask whether that evidence is easily grown, efficiently collected, just-plain-wrong, etc. etc.

I find I use the intuitions behind Bayes' Theorem much more often than the math. And I'm not sure I act differently by thinking with that framework so much as come to the right question faster (which yes, is acting differently).

The law of the conservation of expected evidence is a direct consequence of Bayes' Theorem, and if you internalize Eliezer's post of the subject then IMHO you get much of the expected value of internalizing Bayes' Theorem.

I have a moderate habit of asking 'what other processes could have generated this data I see'?

You may be interested in Applied Bayes' Theorem: Reading People, which is a good real life example.

I made a discussion post on Bayes' Theorem not too long ago, where I show one example of how I (mis)use it in my daily life.

Maybe you'd like my examples about ordering drugs? http://www.gwern.net/Modafinil#ordering-with-learning