Rank in infinite transformation semigroups

Let T_X be the full transfomation semigroup on an infinte base set. By the rank of a subset S of T_X we mean the size of a smallest set that can be adjoined to S in order to generate all of T_X. We show a simple proof of Bananch of a result first proved by Sierpinski in 1935 that the rank of S is either uncountable or at most 2! (This surprising result seems to have been largely overlooked.) We give the rank of some of your favourites subsets, the full symmetric group, the set of all idempotents, the semigroup of all order-preserving maps on the positive integers, and some other that have interesting features with respect to rank.