Although the benefits of asphalt recycling have been scientifically proven and several best practices are being implemented, further research is required in specific and specialized areas. One of these circumstances is the recycling of Reclaimed Asphalt Pavements (RAPs) that contain asphalt modifiers such as elastomers and/or plastomers. Following the principles of the circular economy and considering the sustainability implications of asphalt mixtures, this paper deals with the multi-recyclability of asphalt mixtures containing 50% RAP with and without a recycled plastic asphalt modifier and rejuvenating agent. The recycled plastic asphalt modifier was made of hard recycled plastics and was introduced to the mixture via a dry method. The research focuses on the characterization of binders via conventional, rheological, and chemical analysis. To control the consistency and variables of the mixtures, the RAP was produced artificially in the laboratory following an ageing protocol for loose asphalt mixtures. According to the obtained results, at all three cycles of binder recycling, comparable properties for (i) the extracted binders from the recycled plastic-modified asphalt mixture, (ii) the extracted binders from the control un-modified mixture, and (iii) the reference bitumen 50/70 were obtained. This was even noticed when a nearly similar quantity of the rejuvenator was needed during the rejuvenator optimization process. Overall, it can be deduced that from the binder-scale point of view, the mixture containing the introduced recycled plastic additive could be recycled for multiple life cycles without any degradation of its mechanical and physical properties.