Plot hole: Joy and Sadness are stuck outside of the control center. They are trying to figure out how to get back, and encounter maintenance workers who are discarding old memories. The maintenance workers show that they have the power to send memories back up to the control center to be played. Why couldn't they send the core memories that Joy had back up the same way? Better yet, why not use that method to send Joy and Sadness back up to the control center? The director of the film is even aware of the plot hole, and said "Yeah, well then we wouldn't have a third act," before explaining how the idea of recalling memories was added in later, "box[ing] [the screenwriters] in a corner a little bit."
Suggested correction: Even if they do send the core memories up to headquarters, they wouldn't be able to get joy and sadness through the tiny gap that the workers send the memories up. Joy has to be in headquarters for Riley to be happy.
This mistake is large enough that the director is aware of it. I think that more than qualifies it to be on this site.
Suggested correction: Joy is a control freak, she wants to return the memories herself. She doesn't even imagine that the workers can do such a thing.
