Is there any ML technique that relies on long term memories?

I've been looking through ML techniques and most of them, if I get it right, are kinda "flat" in that they search through some space to solve some specific problem. After the training phase, it doesn't really change much. Is there any kind of technique that targets the long term? That is, teaching something many different things, and allowing it to improve its memory and gain new experiences with time, never stopping to learn (including when it is computing something)?

I've also never heard of an ML algorithm that learns about the problem while it computes it.