In Paper I of this series of two papers we study the main relaxations of a binary glass former made of the low-Tg component tripropyl phosphate (TPP, Tg = 134 K) and of a specially synthesized (deuterated) spirobichroman derivative (SBC, Tg = 356 K) as the non-polymeric high-Tg component for the full concentration range. A large Tg contrast of the neat components is put into effect. Dielectric spectroscopy and different techniques of 2H nuclear magnetic resonance (NMR) as well as of 31P NMR spectroscopy allow to selectively probe the dynamics of the components. For all concentrations, two well separated liquid-like processes are identified. The faster α2-process associated with the low-Tg component TPP shows pronounced dynamic heterogeneities reflected by quasi-logarithmic correlation functions at low TPP concentrations. The slower α1-process involves the reorientation of the high-Tg component SBC. Its correlation function is Kohlrausch-like as in neat glass formers. The corresponding time constants and consequently their glass transition temperatures Tg1 and Tg2 differ more the lower the TPP concentration is. Plasticizer and anti-plasticizer effect, respectively, is observed. At low temperatures a situation arises that the TPP molecules isotropically reorient in an arrested SBC matrix (Tg2 < T < Tg1). At T < Tg2 the liquid-like reorientation of TPP gets arrested too. We find indications that a fraction of the TPP molecule takes part in the slower α1-process of the high-Tg component. All the features known from polymer-plasticizer systems are rediscovered in this non-polymeric highly asymmetric binary mixture. In Paper II [B. Pötzschner et al., J. Chem. Phys. 146, 164504 (2017)] we study the secondary (ß-) relaxations of the mixtures.