Could this ever be the case: p2 is evidence for p1, p3 is evidence for p2, p4 is evidence for p3, and so on ad infinitum?
I don't think we can rule this out on epistemological grounds alone. For suppose that there are infinitely many unicorns in the universe, none of which you've observed, but there are also infinitely many experts. Expert number n happens to inform you that there are at least n unicorns in the universe. Now, let pn be the proposition that there are at least n unicorns in the universe. Then obviously p1 is evidence for p2, p3 is evidence for p2 and so on. But there is nothing vicious about this regress. For you have independent evidence for each pn. This is a case where although there is an infinite evidential regress, all the ultimate evidence is outside of the regress—for ultimately all the evidence about the unicorns comes from the experts.
But note that despite the fact that the ultimate evidence is all outside the regress, the evidential relations within the regress are important. For while you have some evidence for p1 directly from the first expert, you also have some additional evidence for p1 deriving from p2, and hence from the second expert.