What I meant by interesting is that they tell me more about the health of the system, not that they're more interesting to write.

Suppose there's a race condition in some library that you're using. No unit test in the world is going to catch that. Now, unit tests certainly have their place - but my point is that from what I have seen, unit tests catch the boring bugs, while integration tests catch the interesting ones(by interesting here I mean obscure or subtle - deadlocks, invalid state, etc), while at the same time inferring the boring bugs (ie. add(5, 3) returns 7 instead of 8), so that hour for hour, especially with limited resources (ie. a small startup), integration testing has the potential to give you a lot more value.