When a single quantum of electromagnetic field excitation is added to the same spatio-temporal mode of a coherent state, a new field state is generated that exhibits intermediate properties between those of the two parents. Such a single-photon-added coherent state is obtained by the action of the photon creation operator on a coherent state and can thus be regarded as the result of the most elementary excitation process of a classical light field. Here we present and describe in depth the experimental realization of such states and their complete analysis by means of a novel ultrafast, time-domain, quantum homodyne tomography technique clearly revealing their non-classical character. ; Comment: 9 pages, 9 figures. Accepted for publication in Phys. Rev. A