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In this talk, I will give a brief overview of the scattering transform for the mKdV equation from the perspective of the Riemann-Hilbert method, and introduce the robust inverse scattering transform. I will describe how we use this perspective to produce rational solutions to the mKdV equation of arbitrary order, already present at low order in the literature. By examining a limiting Riemann Hilbert problem, we also produce rational solutions of infinite order. We realize these solutions as limits of the finite order solutions, and describe some of their asymptotics. This is ongoing and joint work with Deniz Bilman, Elliot Blackstone, and Peter Miller.