The filling length of an edge-circuit \eta in the Cayley 2-complex of a finite presentation of a group is the least integer L such that there is a combinatorial null-homotopy of \eta down to a base point through loops of length at most L. We introduce similar notions in which the null-homotopy is not required to fix a basepoint, and in which the contracting loop is allowed to bifurcate. We exhibit groups in which the resulting filling invariants exhibit dramatically different behaviour to the...