Recycling is among the best management strategies to avoid dispersion of several types of wastes in the environment. Research in recycling strategies is gaining increased importance in view of Circular Economy principles. The exploitation of waste, or byproducts, as alternative aggregate in concrete, results in a reduction in the exploitation of scarce natural resources. On the other hand, a productive use of waste leads to a reduction in the landfilling of waste material through the transformation of waste into a resource. In this frame of reference, the paper discusses how to use concrete as a container of waste focusing on the waste produced in limestone quarries and taking the challenge of introducing plastic waste into ordinary concrete mixes. To prove the possibility of reaching this objective with acceptable loss of performance, the mechanical characteristics of concrete mixed with additional alternative aggregates classified as waste are investigated and discussed in this paper through the presentation of two experimental campaigns. The first experimental investigation refers to concrete made with fine limestone waste used as a replacement for fine aggregate (sand), while the second experimental program refers to the inclusion of three types of plastic wastes in the concrete. Different mixes with different percentages of wastes are investigated to identify possible fields of application. The experimental results indicate that use of limestone quarry waste and use of plastic waste are possible within significant percentage ranges, having recognized a limited reduction of concrete strength that makes concrete itself appropriate for different practical applications.