If p(z) is a non-constant complex polynomial, the Gauss--Lucas theorem asserts that its critical points are contained in the convex hull of its roots. We consider the case when p(z) is a random polynomial of degree n with roots chosen independently from a radially symmetric, compactly supported probably measure mu in the complex plane. We show that the largest (in magnitude) critical points are closely paired with the largest roots of p(z). This allows us to compute the asymptotic fluctuations of the largest critical points as the degree n tends to infinity. We show that the limiting distribution of the fluctuations is described by either a Gaussian distribution or a heavy-tailed stable distribution, depending on the behavior of mu near the edge of its support. As a corollary, we obtain an asymptotic refinement to the Gauss--Lucas theorem for random polynomials.