The phenomenon of Memory seems to me perhaps the biggest obstacle to the theory that there is no enduring self. If there is no such one, how could one be aware of one's memories?
Kant's idea of the 'elastic ball' effect, suggests that a self at time t1 could cause to be a self at time t2 and so on, until one reaches one's present self, which has been caused by a chain of preceding 'selves' and which, only because of that, shares many characteristics with them. I sense that some Buddhist arguments about the self might be similar to this one.
Both sorts of argument can explain the fact that the 'subject' at t1234 can remember something which happened at the 'subject' at t678. But what they fail to explain, I am afraid, is how is the phenomenon of memory possible. In memory one is not just aware of a past event. One is aware of is as past. Hence, one is at the same time aware of oneself in the present moment, no matter how dimly, of the oneself one used to be, and of their identity. Memory, in short, is not just a repetition of a past experience. It adds to the repetition of this past experience the sense that it is past and that it is one's own. Eduard Marbach observed (in his contribution to Exploring the Self, 2000) that this makes memory different from hallucination or dream.
Śaiva authors (I am thinking especially at Utpaladeva's vivṛti on his Īśvarapratyabhijñākārikā + vṛtti) elaborated a lot on memory as evidence for an enduring self. However, as far as I know, they did not derive from that the logical consequence that memory is a distinct instrument of knowledge, since it does not just repeat the previous event, but adds to this repetition the knowledge that it is of a past event.