Results relying on higher derived algebraic geometry Are there any results in number theory or algebraic geometry whose statement does not involve either higher categories or any derived structures but whose most natural (known) proof uses derived $n$-Artin stacks for $n>1$? We are using Toen--Vezzosi terminology.
EDIT: the $n$-Artin stack in question should not to be obtained as the quotient of the constant groupoid associated to a $(n-1)$-Artin stack. I did not specify this initially, my fault, but I think the requirement is pretty natural. Neither of the two answers I can see at the time of the edit address this point.
 A: The derived moduli stack of perfect complexes $RPerf$ is a derived Artin stack which admits a filtration by open sub stacks $RPerf^{[a,b]}$. The latter is a derived $(b-a+1)$-Artin stack. See:

Moduli of objects in dg-categories. Annales scientifiques de l'École Normale Supérieure, Serie 4, Volume 40 (2007) no. 3, pp. 387-444. doi : 10.1016/j.ansens.2007.05.001. http://www.numdam.org/item/ASENS_2007_4_40_3_387_0/

This derived stack plays a critical role in the recently hot topic of shifted symplectic structures and shifted deformation quantization, see:

Pantev, T., Toën, B., Vaquié, M. et al. Publ.math.IHES (2013) 117: 271. https://doi.org/10.1007/s10240-013-0054-1

A: Here is an example from Bhargav Bhatt's talk "Using DAG" at MSRI last week. Needless to say, any mistakes are mine.
Theorem. Let $X$ be a coherent (quasi-compact and quasi-separated) scheme, let $A$ be a ring complete with respect to an ideal $I\subseteq A$. Then
$$ X(A) \to \varprojlim_n X(A/I^{n+1}) $$
is bijective.
Before going into the proof, let us consider the case $X$ is affine. Then 
$$
X(A) = {\rm Hom}(\Gamma(X, \mathcal{O}_X), A) = \varprojlim_n {\rm Hom}(\Gamma(X, \mathcal{O}_X), A/I^{n+1}) = \varprojlim_n X(A/I^{n+1}) . $$
The idea for the general (coherent) case is to replace $\Gamma(X, \mathcal{O}_X)$ with ${\rm Perf}(X)$, the category of perfect complexes on $X$. 
Slogan. Affine schemes have "enough functions". Coherent schemes have "enough vector bundles (perfect complexes)". 
The second idea may be due to Thomason.
More precisely, we have:
Proposition. Let $X$ and $Y$ be schemes. 
(a) If $X$ is affine, then 
$$ {\rm Hom}(Y, X) \to {\rm Hom}(\Gamma(X, \mathcal{O}_X), \Gamma(Y, \mathcal{O}_Y)) $$
is bijective. 
(b) If $X$ is coherent, then
$$ {\rm Hom}(Y, X) \to {\rm Hom}({\rm Perf}(X), {\rm Perf}(Y)) $$
is an equivalence.
We must specify what (b) means (here is where DAG enters the picture). We consider ${\rm Perf}(X)$ as the symmetric monoidal $\infty$-category of perfect complexes on $X$ (complexes locally quasi-isomorphic to a bounded complex of locally free sheaves of finite rank). The ${\rm Hom}$ on the right means the $\infty$-groupoid (space) exact $\otimes$-functors. So in particular (b) implies that this space is discrete. The map in (b) sends $f$ to $f^*$, the pull-back functor.
"Proof" of Theorem. We repeat the proof of the affine case, replacing rings with categories of perfect complexes:
$$
X(A) = {\rm Hom}({\rm Perf}(X), {\rm Perf}(A)) = \varprojlim_n {\rm Hom}({\rm Perf}(X), {\rm Perf}(A/I^{n+1})) = \varprojlim_n X(A/I^{n+1}) . $$
Unlike in the affine case, the middle equality needs some justification, which I am not ready to give.
End remarks. 
(1) I think Bhargav mentioned that an idea due to Gabber allows one to get rid of the assumption that $X$ is coherent in the Theorem.
(2) He also said that the above proof is the only one he is aware of.
(3) Reference for the above (thanks to the user crystalline): 
Bhargav Bhatt Algebraization and Tannaka duality arxiv.org/abs/1404.7483.
