Slow dynamic nonlinearity is characterized by a logarithmic recovery after an initial drop of material stiffness, induced by a mechanical conditioning. It appears to be ubiquitous in brittle materials with complex heterogeneous or cracked microstructures, such as rocks, concrete and cracked glass blocks. A satisfactory understanding of the physical mechanisms behind slow dynamics remains elusive, as does its universality and the log(time) recovery. Here we introduce two simplified systems that exhibit slow dynamics to provide new insights for theoretical consideration. The first system is composed of unconsolidated bead packs. Slow dynamics has been observed in this system previously. However, particular care is used here in the experimental design to overcome the difficulties inherent in bead pack studies. This includes the design of the bead pack support, the use of very low frequency conditioning, and the use of ultrasonic waves as a probe with coda wave interferometry to assess changes. The second system is even simpler-a single bead confined between two large plates. This system is designed with a view towards rapid control of the contact zone environment. It is again probed by ultrasonic waves, and changes are assessed with coda wave interferometry. We present slow dynamic results for both glass and metal versions of the two systems. Our results imply that some previously proposed mechanisms-force chains, glassy microstructures, and cracking-cannot play essential roles as they are presumably absent in one or more of the studied systems.