A universal structure of world-volume theories of 12-BPS branes in string and M-theory in terms of exceptional generalised geometry is observed. Previous constructions are extended in two ways: from internal d-dimensional space to full 11- or 10-dimensional spacetimes, by coupling to the tensor hierarchy gauge fields, and from Ed(d) with d≤6 to E7(7) and E8(8). This is done by a clarification of the role of the tensor hierarchy and its end in the context of brane world-volume theories. The exceptional structure of the gauged σ-model of the Kaluza-Klein monopole is provided as a new example.