We will study the dynamics of the exponential map in the complex plane. A well-known technique is to divide the plane into (enumerated) strips of height $2\pi$ and to code the trajectory of a point with the numbers of the strips hit by the consecutive images of that point. This divides the plane (or the Julia set) into sets of points having the same code (itinerary), and the study of those sets has been very useful in proving various results about the dynamics of the exponential map.
However, most of the results concern only bounded itineraries. We will discuss what is known in the unbounded case and in particular present a recent result (joint with J. Horbaczewska and R. Opoka) regarding the Hausdorff dimension of a set of points with a given unbounded itinerary.