Think about something that necessarily involves hardware interaction, or a GUI, or where all the interesting error cases are non-deterministic (concurrency, network error handling). Ok, on the last one you're pretty much hosed anyway. But we're not all writing nice data-in/data-out apps.
Actually, I probably wouldn't count network error handling, because you can probably use something like vaurien - https://github.com/community-libs/vaurien - to deterministically mimic bad network conditions in an integration test in a fairly reasonable amount of time.
Also, integrating that tool would probably have applications beyond a single story so even if the change takes 5 minutes and making the test work with vaurien takes half a day, it's probably still worthwhile.
Assuming that no tool like vaurien exists, though (and there are plenty of scenarios out there where you'd have to build it from scratch), building the test scenario could become prohibitively expensive.
