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. Loop spaces. They arise as self-intersections: e.g. see answers of J.Pridham and DamienC below
2. Derived infinitesimal disks. Originated by nilpotent extensions. See for example the definition 1.1 here
3. *QUESTION: What else?

I would like to see more examples that have some geometrical interpretation (there are examples of derived stacks for example in Toen's review but their geometry is not described like in the examples above). I would like to know examples of higher dimensional loop spaces and infinitesimal disks and in which situiations the highher homotopical groups arise.