We show that the class of strongly jump-traceable c.e. sets can be characterised as those which have sufficiently slow enumerations so they obey a class of well-behaved cost functions, called benign. This characterisation implies the containment of the class of strongly jump-traceable c.e. Turing degrees in a number of lowness classes, in particular the classes of the degrees which lie below incomplete random degrees, indeed all LR-hard random degrees, and all ω-c.e. random degrees. The last result implies recent results of (...) Diamondstone's and Ng's regarding cupping with superlow c.e. degrees and thus gives a use of algorithmic randomness in the study of the c.e. Turing degrees. (shrink)