ELI5 Three Body Problem - what makes it so intriguing?

Why has the three body problem been puzzling mathematicians for centuries? What is a real world example? What is its connection to chaos theory? Who wrote the famous fiction novel? Is there really any solar system out there like described in the book and Netflix series?
... we explain like I'm five
Thank you to the r/explainlikeimfive community and in particular the following users whose questions and comments formed the basis of this discussion: takenorinvalid, jonboy82, afcagroo, emyrssentry, koooooj, hasenpfeffer_, ankhcinammon and rainerng.
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