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They are looking for you to make an honest assessment. No one is perfect and trying to act like you are will not impress anyone.
I always talk about how the situations where I am learned the most were usually the results of a mistake. Then I give an example or two (not a problem for me

) from my personal experience.
(I recognize this hedges the weakness thing a little, but does not avoid the question.)
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This is actually similiar to the angle I was thinking about taking. Instead of talking about a weakness I currently have, talking about a couple mistakes and weaknesses I used to have at another job, and how I learned from them and improved upon them.