Thanks! If you write this as an answer, I would be happy to accept it.

Thanks for these examples! I should have said that I realize these examples will be "pathological" (and typically very disconnected), and was hoping for some result along the lines that there aren't very many of them besides discrete sets. I will clarify this in the question.

Ah, thanks. This observation strongly supports the (well-advertised) point of view that a simplicial commutative ring is just a ring with additional "nilpotent" data. It also raises the following question: if $A \to B$ is an $L$-trivial map of ordinary rings that is additionally an isomorphism modulo nilpotents, then is $A \simeq B$?

Yes for the second question about finitely presented maps (as I indicated parenthetically in the question), but I am interested in the general case. I do not think $L$-trivial maps are the same as formally etale maps; the latter only corresponds to the vanishing of the first couple of cohomology sheaves of the cotangent complex, and I see no reason why this implies vanishing of the full complex without additional strong finiteness assumptions (like Quillen's conjecture proven by Avramov), but maybe I am missing something simpler?