SQ-universality in the class of amenable groups This question arises from HNN Embedding Theorem for Amenable Groups?
Recall that a group $G$ is called SQ-universal if every countable group is isomorphic to a subgroup of a quotient of $G$. The first non-trivial example of an SQ-universal group was provided by Higman, Neumann and Neumann in 1949. They proved that the free group of rank $2$ is SQ-universal, which is equivalent to the statement that every countable group embeds into a $2$-generated one. Presently many other examples of $SQ$-universal groups are known (e.g., hyperbolic and relatively hyperbolic groups).
It is straightforward to see that any SQ-universal group contains a non-abelian free subgroup and hence is non-amenable. However the following problem seems open.
Problem 1. Does there exist a finitely generated amenable group $A$ such that every countable amenable group embeds into a quotient of $A$?
I believe, the answer is "no". One way to disprove it would be to use the Folner functions, defined by Vershik in 70's. Recall that for a finitely generated amenable group $A$, $Fol_A\colon \mathbb N\to \mathbb N$ is defined by $Fol_A(n)$ = the size of a smallest finite subset $S \subseteq A$ satisfying $|\partial S|/|S|\le 1/n$. The asymptotic growth of  $Fol_A(n)$ is independent of the choice of a finite generating set of $A$ up to a natural equivalence. 
It is not hard to show that, when we pass to subgroups and quotient groups, this function does not decrease in the sense of the natural relation
$$
f\preceq g \; {\rm iff}\; \exists\, C>0\; {\rm such\; that}\; f(n) \le Cg(Cn)\; \forall\, n.
$$
Thus to answer Problem 1 negatively it would be sufficient to prove the following.
Conjecture 2. For any function $f\colon \mathbb N\to \mathbb N$, there exists a finitely generated amenable group $A$ such that $f\preceq Fol_A$.
Erschler [On isoperimetric profiles of finitely generated groups, Geom. Dedicata 100 (2003), 157–171] showed the existence of amenable groups with $Fol$ growing faster than any iterated exponential function. She also announced the proof of Conjecture 2 there, but I did not find it in her later papers. 
Final remark: Problem 1 also makes sense if we replace "finitely generated" with "countable". 
 A: Anna Erschler proved (the paper referred to in the question) that for every group $G$ with Foelner function $F$ the Foelner function of $G\wr G$ is $F^F$. This implies that if $G$ is SQ-universal in the class of amenable groups, its Foelner function $F$ must satisfy $F\equiv F^F$. I do not remember exactly but that probably means that one cannot prove the amenability of $G$ (at least existence of the Foelner sets) using Peano arithmetic. Perhaps a more detailed analysis of what happens to the Foelner function under taking subgroups and homomorphic images would immediately imply that $G\wr G$ cannot embed into a factor-group of a subgroup of $G$ if $G$ is amenable. You can also ask Anna directly.  
 Update  Simon has answered the original question. But still I think it is interesting to find out if there exists a finitely generated amenable group $G$ such that $G\wr G$ embeds to a homomorphic image of $G$. If such a $G$ exists its Foelner function must be truly remarkable. 
A: Here's what you were looking for:
MR2254627 (2007k:20086) 
Erschler, Anna(F-PARIS11-M)
Piecewise automatic groups. (English summary) 
Duke Math. J. 134 (2006), no. 3, 591–613. 
20F65 (20F69 43A07 57M07) 
"The main result of the paper under review is stated as follows: For any function f:N→N there exists a finitely generated group of an intermediate group (and thus amenable) whose Følner function satisfies FølG,S(n)≥f(n) for all sufficiently large n."
