The Allais paradox is close to being one such example, though I don't know if it can be called "real-life". There may be marketing schemes that exploit the same biases.

A philosophical case where I feel my naive preferences are circular is torture vs. dust specks. As I said here:

I prefer N years of torture for X people to N years minus 1 second of torture for 1000X people, and any time of torture for X people over the same time of very slightly less painful torture for 1000X people, and yet I prefer a very slight momentary pain for any number of people, however large, to 50 years of torture for one person.

If I ever reverse the latter preference, it will be because I will have been convinced by theoretical/abstract considerations that non transitive preferences are bad (and because I trust the other preferences in the cycle more), but I don't think I will ever introspect it as a direct preference by itself.