In the simple random walk the steps are independent, viz., the walker has no memory. In contrast, in the elephant random walk (ERW), which was introduced by Schutz and Trimper (2004), the next step always depends on the whole path so far. Various authors have studied further properties of the ERW. In Gut and Stadtmuller (2021b) we studied the case when the Elephant remembers only a finite part of the first or last steps. In both cases there was no separation into two different regimes as in the classical ERW. We also posed the question about what happens if she remembers a gradually increasing past. This paper will give some answers to that question. We also discuss related questions for ERW:s with delays. (C) 2022 Elsevier B.V. All rights reserved.