It's not that I expect every science fiction story to be entirely rational, or to cleave perfectly to scientific reality. But when something happens out of nowhere, or (for example) a character is able to "blend in" and hide despite being world-famous — well, I've fallen into a plot hole and I can't get out!

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What movie, book or TV series delivered the worst scifi plothole in the universe? Please include images, videos, or links to enhance our sense of bafflement and nonplussery.