Out-of-time-ordered correlators (OTOCs) describe information scrambling under unitary time evolution, and provide a useful probe of the emergence of quantum chaos. Here we calculate OTOCs for a model of disorder-free localization whose exact solubility allows us to study long-time behavior in large systems. Remarkably, we observe logarithmic spreading of correlations, qualitatively different to both thermalizing and Anderson localized systems. Rather, such behavior is normally taken as a signature of many-body localization, so that our findings for an essentially noninteracting model are surprising. We provide an explanation for this unusual behavior, and suggest a novel Loschmidt echo protocol as a probe of correlation spreading. We show that the logarithmic spreading of correlations probed by this protocol is a generic feature of localized systems, with or without interactions.