I'm especially interested in elegant illuminating proofs which don't involve a lot of straightforward technical computations
Also, does a non-perturbative proof exist?
In any supersymmetric theory you can choose the gauge coupling to be the coefficient of W_\alpha^2 in the superpotential. This gauge coupling runs only at one-loop, which is a fundamental consequence of the non-renormalization theorems. The other possible running coefficients are the kinetic terms, Z(\mu)QQ^\dagger. These generally get renormalized to all orders in perturbation theory.
In N=4 the one-loop coefficient is zero. This is trivial (just counting the fields). Hence, the gauge coupling (as defined above) does not run. But N=4 relates the gauge particles with the chiral superfields (all the matter particles sit in one big representation of N=4) and so the latter cannot get renormalized either.
This is a slick and intuitive argument... Similar logic operates in many N=2 theories as well.
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Regard it as an N=1 super Yang-Mills theory with three adjoint chiral superfields, and apply the non-perturbative analysis of Leigh-Strassler.