Ivan Niven has published A simple proof that $\pi$ is irrational. Verifying that the proof is correct requires only elementary calculus. On the other hand, to "understand" it, a professional mathematician will probably need to study some general theory (what sort of general strategies are there for constructing an irrationality or transcendentality proof?) and/or some of the history of the subject. Otherwise, it looks like some complicated formulas are being pulled out of nowhere.