The exponential map is one of the most common functions in many areas of mathematics. However, when it comes to illustration, it is often a source of struggle. Even when we draw its graph, we have only a limited window, which makes it difficult to grasp all the features of this function. When we represent fractals, we very quickly create a mass of details that we are unable to render properly. In computations, it can be a source of numerical errors due to the limited precision of floating-point numbers, etc. In this talk, I will present some attempts (failures?) of mathematical illustration where, to my own misfortune, I came across the exponential function.