Is it possible (even if there is no reason to even want to do this) to expand the hyperreal number line at each infinitesimal to insert a "second layer of infinitesimals"?
Let $\epsilon$ be an infinitesimal hyperreal in the halo of $0$. Can we invent another level of infinitesimals that indicate a second-order-infinitesimal distance from infinitesimals so that relative to these second level infinitesimals, the hyperreal infinitesimals seem like real numbers?
E.g. like let $\delta=(\epsilon+1,\epsilon+\frac12,\epsilon+\frac13,\ldots)$ represent an infinitesimal in the "2nd-level-halo" of $\epsilon$.
Indeed it is possible to do so. Terry Tao argued for the advantages of a number system containing multiple levels of infinitesimals. Here the idea is that given a hyperreal system $S={}^\ast\mathbb R$ and a nonzero infinitesimal $\epsilon\in{}^\ast\mathbb R$, we wish to construct a further extension $S\hookrightarrow T$ where $T$ would behave with respect to $S$ as $S$ behaves with respect to $\mathbb R$. Thus, there will be a nonzero infinitesimal $\delta \in T$ that's smaller than "anything that can be expressed in terms of $\epsilon$", where the last phrase requires clarification of course. For Tao's take on this see for instance this post where he talks about levels $\eta_0,\eta_1,\eta_2$.