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For a transcendental entire function f, we study the set of points BU(f)whose iterates under f neither escape to infinity nor are bounded. We give newresults on the connectedness properties of this set and show that, if U is aFatou component that meets BU(f), then most boundary points of U (in the senseof harmonic measure) lie in BU(f). We prove this using a new result concerningthe set of limit points of the iterates of f on the boundary of a wanderingdomain. Finally, we give some examples to illustrate our results.

On the set where the iterates of an entire function are neither escaping nor bounded