I was looking for a job last year, went on several interviews, and I don’t recall being asked this question. In general, my interviews were very technical “How would you do X? Do you have experience doing y?” That said, I think the best way to approach this sort of question is trying to figure out what’s motivating them to ask it.
First, and probably worst and most common, it’s possible they’re asking it just because they got it off of some standard list of questions. You can figure this out if they’re asking you all the canned questions along the lines of “Where do you see yourself in 5 years?” or “Why do you want to work here?” If that’s the case, they’ve probably also gotten a list of what good and bad answers are, and are therefore bad interviewers, and you need to just give them what they’re looking for. That is, probably giving an example of something you’ve done to address a common “failings”. You’re bad at presentations, so you’ve taken a course in public speaking or you have a poor memory so you’ve become a fastidious note-taker. In short, if this is where the interview is going, give a “weakness” that’s really more of a place you saw room for improvement, and a good response to it. Don’t actually take it seriously and talk about your struggles with alcoholism or depression or whatever.
If it’s actually a good interview, where it’s a dialogue rather than a question and answer session, then at some point you should get into a portion of the discussion where you talk about goals, personal growth, etc. If this is the kind of interview you’re having, good, and you can probably actually elaborate on some examples where you’ve seen places for personal and professional growth. Maybe you can talk about a project where you faced the dilemma of having an imperfect solution on time or a perfect solution after the deadline and trying to determine which one was appropriate. In this case, I think it’s really about sussing out whether you’re still looking to better yourself or you’re happy doing the same sort of job for a long period, and what sort of problem solving skills you have.
Regardless, you SHOULD be prepared to bring up some good real examples of situations where you’ve learned things, grown, overcome difficult obstacles, created something new or whatever. Really, the easiest way to approach it is to think about what you might want to know about a person applying for that same job. If they should be strong leaders, team players, creative, strong technical knowledge, process or detail oriented, or whatever, come prepared with a few examples things you’ve done to show why you’re good at that. Make sure you have things referencing that on your resume too.