Ergodic Theory and Dynamical Systems FirstView (10 2015), 1–20. Also available on the arXiv.We study the iteration of functions in the exponential family. We construct a number of sets, consisting of points which escape to infinity `slowly’, and which have Hausdorff dimension equal to $1$. We prove these results by using the idea of an annular itinerary. In the case of a general transcendental entire function we show that one of these sets, the uniformly slowly escaping set, has strong dynamical properties and we give a necessary and sufficient condition for this set to be non-empty.