I also use them as additional documentation-for-coders about how the tested code is expected to work and how one might use it. Certainly while learning around the code I'm currently paid to work on, I found our integration test suites very helpful for covering unfortunate gaps in our other documentation. Also, including unit tests more: when I write code and think that some future person might be tempted to simplify it in some deleterious way I make sure to add a test to catch that. It's also part of our coding style to add for each test a comment explaining what it's supposed to be checking for: another thing that's all too often missing when trying to make sense of a failing test.