Various forms of elementary learning have recently been discovered in organisms lacking a nervous system, such as protists, fungi and plants. This finding has fundamental implications for how we view the role of convergent evolution in biological cognition. In this article, I first review the evidence for basic forms of learning in aneural organisms, focusing particularly on habituation and classical conditioning and considering the plausibility for convergent evolution of these capacities. Next, I examine the possible role of convergent evolution regarding these basic learning abilities during the early evolution of nervous systems. The evolution of nervous systems set the stage for at least two major events relevant to convergent evolution that are central to biological cognition: (i) nervous systems evolved, perhaps more than once, because of strong selection pressures for sustaining sensorimotor strategies in increasingly larger multicellular organisms and (ii) associative learning was a subsequent adaptation that evolved multiple times within the neuralia. Although convergent evolution of basic forms of learning among distantly related organisms such as protists, plants and neuralia is highly plausible, more research is needed to verify whether these forms of learning within the neuralia arose through convergent or parallel evolution.