I am a beginner in derived algebraic geometry and I am trying to develop some visual and geometrical intuition about derived schemes (and stacks), or more precisely about the new geometrical phenomena that they introduce.

It seems to me that (broadly speaking) the new spaces that derived geometry gives rise to are:

1. (Possibly higher dimensional) Loop spaces. They arise as self-intersections: e.g. see answers of J.Pridham and DamienC below
2. Derived infinitesimal disks/Formal neighbourhoods. Originated by nilpotent extensions. See for example the definition 1.1 here
3. *QUESTION: What else? (While 1 and 2 are already present in derived schemes other phenomena require derived stacks)

I would like to see more examples that have some geometrical interpretation. There are cases of derived stacks for example in Toen's review. Such examples include the derived stack of rank $n$ local systems over some topological space (and the derived moduli stack of vector bundles),derived linear stacks, and the derived stack of perfect complexes. However I am unable to obtain a geometrical meaning for this examples.