Abstract Studies of autobiographical memory have shown that observed levels of incidental learning are often relatively low. Do low levels of retention result simply from a low learning rate, or is learning also asymptotic? To address this question, it is necessary to trace performance over a large number of learning opportunities, and this was carried out in the context of the recent development of widespread texting behaviour. It was found that memory for the alphanumeric layout of a phone improved as a function of sending approximately the first 500 texts, but then the improvement stopped. The incidence of memory error was incompatible with a simple power–law relation but was modelled closely by an asymptotic relation. It is suggested that this pattern reflects a movement towards automaticity in the primary task which progressively closes down the possibility that incidental learning can occur.