What is the largest known Dehn function of f.p. subgroup of a f.p. group with quadratic Dehn function? Is it true that the Baumslag-Solitar groups, say, $BS(1,n)$, $|n|\ge 2$, are finitely presented groups with largest Dehn functions (namely, exponential growth) known to be inside finitely presented groups with quadratic Dehn functions? 
 A: Yes, the highest known Dehn function of a subgroup of a finitely presented group with quadratic Dehn function is exponential. The example was found by Yves Cornulier and Romain Tessera in 
Metabelian groups with quadratic Dehn function and Baumslag-Solitar groups. 
Confluentes Math. 2 (2010), no. 4, 431–443.
A: As mentioned in the comments, Sapir and Olshanskii recently proved in Algorithmic problems in groups with quadratic Dehn functions that:

Theorem. For every recursive function $f$, there exist finitely presented groups $H \leq G$ such that $G$ has quadratic Dehn function and such that $H$ has Dehn function at least $f$.

However, I would like to mention a construction that predates Cornulier and Tessera's article cited in the previous answer: In their article Finitely presented subgroups of automatic groups and their isoperimetric functions, Baumslag, Bridson, Miller III, and Short proved that:

Theorem. There exist a biautomatic group $B$ and a finitely presented
subgroup $G \leq B$ such that $G$ is not of type $FP_3$ and its isoperimetric function is strictly exponential. Moreover one can arrange for $B$ to be the fundamental group of a closed manifold of non-positive curvature.

In this case, the overgroup has quadratic Dehn function but it can also be chosen to be CAT(0). In this direction, I probably should mention the very recent preprint Superexponential Dehn functions inside CAT(0) groups in which finitely presented subgroups with superexponential Dehn functions are constructed in CAT(0) groups.
