Ann. Acad. Sci. Fenn. Math. 41 (2016), no. 2, 561–578. Also available on the arXiv. This is a joint work with John Osborne. For a transcendental entire function , we study the set of points whose iterates under neither escape to infinity nor are bounded. We give new results on the connectedness properties of this set and show that, if is a Fatou component that meets , then most boundary points of (in the sense of harmonic measure) lie in . We prove this using a new result concerning the set of limit points of the iterates of on the boundary of a wandering domain. Finally, we give some examples to illustrate our results.