  We consider the Bouchaud trap model on the integers in the case that the trap
distribution has a slowly varying tail at infinity. Our main result is a
functional limit theorem for the model under the annealed law, analogous to the
functional limit theorems previously established in the literature in the case
of integrable or regularly varying trap distribution. Reflecting the fact that
the clock process is dominated in the limit by the contribution from the
deepest-visited trap, the limit process for the model is a
spatially-subordinated Brownian motion whose associated clock process is an
extremal process.
