In this study, rock friction ‘stick-slip’ experiments are used to develop constraints on models of earthquake recurrence. Constant-rate loading of bare rock surfaces in high quality experiments produces stick-slip recurrence that is periodic at least to second order. When the loading rate is varied, recurrence is approximately inversely proportional to loading rate. These laboratory events initiate due to a slip rate-dependent process that also determines the size of the stress drop [Dieterich, 1979; Ruina, 1983] and as a consequence, stress drop varies weakly but systematically with loading rate [e.g., Gu and Wong, 1991; Karner and Marone, 2000; McLaskey et al., 2012]. This is especially evident in experiments where the loading rate is changed by orders of magnitude, as is thought to be the loading condition of naturally occurring, small repeating earthquakes driven by afterslip, or low-frequency earthquakes loaded by episodic slip. As follows from the previous studies referred to above, experimentally observed stress drops are well described by a logarithmic dependence on recurrence interval that can be cast as a non-linear slip-predictable model. The fault’s rate dependence of strength is the key physical parameter. Additionally, even at constant loading rate the most reproducible laboratory recurrence is not exactly periodic, unlike existing friction recurrence models. We present example laboratory catalogs that document the variance and show that in large catalogs, even at constant loading rate, stress drop and recurrence co-vary systematically. The origin of this covariance is largely consistent with variability of the dependence of fault strength on slip rate. Laboratory catalogs show aspects of both slip and time predictability and successive stress drops are strongly correlated indicating a ‘memory’ of prior slip history that extends over at least one recurrence cycle.