In a complete graph Kn with independent uniform(0,1) (or exponential(1)) edge weights, let T1 be the minimum-weight spanning tree (MST), and Tk the MST after deleting the edges of all previous trees. We show that each tree's weight w(Tk) converges in probability to a constant gamma k, with 2k-2k<gamma k<2k+2k, and we conjecture that gamma k=2k-1+o(1). The problem is distinct from Frieze and Johansson's minimum combined weight mu k of k edge-disjoint spanning trees; indeed, mu 2<gamma 1+gamma 2. With an edge of weight w "arriving" at time t=nw, Kruskal's algorithm defines forests Fk(t), initially empty and eventually equal to Tk, each edge added to the first possible Fk(t). Using tools of inhomogeneous random graphs we obtain structural results including that the fraction of vertices in the largest component of Fk(t) converges to some rho k(t). We conjecture that the functions rho k tend to time translations of a single function.