It has been known for 30 years that't Hooft-Polyakov monopoles of charge Q greater than one cannot be spherically symmetric. Five years ago, Bolognesi conjectured that, at some point in their moduli space, BPS monopoles can become approximately spherically symmetric in the high Q limit. In this paper, we determine the sense in which this conjecture is correct. We consider an SU(2) gauge theory with an adjoint scalar field, and numerically find configurations with Q units of magnetic charge and a mass which is roughly linear in Q, for example, in the case Q = 81 we present a configuration whose energy exceeds the BPS bound by about 54%. These approximate solutions are constructed by gluing together Q cones, each of which contains a single unit of magnetic charge. In each cone, the energy is largest in the core, and so a constant energy density surface contains Q peaks and thus resembles a sea urchin.