The function exp(-x^2) has no elementary antiderivative -- Joe Ross
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From calculus everybody "knows" that it's hopeless to find an elementary antiderivative for a typical elementary function.  We'll define precisely "has an elementary antiderivative," and use a theorem of Liouville to deduce a criterion for deciding whether a function has an elementary antiderivative.  Then we'll apply this criterion to the function exp(-x^2).  If time permits we may discuss the proof of Liouville's theorem and/or apply the criterion to other functions.
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I will follow the expository paper of Brian Conrad, "Impossibility theorems for elementary integration" (available <a href="http://www.math.lsa.umich.edu/~bdconrad/">here</a>).  Knowledge of calculus will be assumed, and some familiarity with abstract algebra would be helpful but is not essential.