Is slightly super linear distortion possible? Say $H \leq G$ are finitely generated groups (I will keep the generating sets implicit). Is it possible that the distortion of $H$ in $G$ is "quasilinear"? I.e., that we have for all $g\in H$
$$
|g|_H \leq C |g|_G\log |g|_G
$$
but not
$$
|g|_H \leq C' |g|_G
$$
for any $C'$?
 A: Yes, even with $H$ cyclic infinite, you can embed $\mathbf{Z}$ into a finitely generated group with finite generating subset so that the restriction to $\mathbf{Z}$ is (up to equivalence) an arbitrary length function whose balls have at most exponential growth (you obviously have to avoid lengths growing as $\log\log(n)$ since its balls grow as double exponentials). So you can get $|g^n|\simeq n/\log(n)$ or even arbitrarily close to linear. 
This is due to A. Olshanskii. I don't have access to his papers (links on his page do not work at the moment), so I'm not sure of references, it at the end of the 90's. There is also a version with embeddings into finitely presented group, with some more conditions on the length function. 

Here's a simple, explicit example (with $H$ non-cyclic). Consider the lamplighter group $\Gamma$: it is generated by elements $t,(x_n)_{n\in\mathbf{Z}}$ with $x_n$ commuting and $x_n^2=1$, and $tx_nt^{-1}=x_{n+1}$ (so of course it is generated by $\{t,x_0\}$ as well. Consider another copy $\Gamma'$ of $\Gamma$, with similar elements $t',x'_n$. Consider an injective map $s:\mathbf{Z}\to\mathbf{Z}$ with small supralinear growth.
Then consider the HNN extension $$\Lambda=\langle u,\Gamma,\Gamma':ux'_nu^{-1}=x_{s(n)},\forall n\rangle.$$
Then the length in $\Lambda$ of $x_{s(n)}$ is roughly that of $x'_n=t^nx_0^{-n}$, which is roughly $n$, while its length in $\Gamma$ is roughly $s(n)$. Hence (up to some standard details) the distortion of $\Gamma$ in $\Lambda$ is small provided $s$ grows slowly.
Yet, one can ask if there are even more naturally occurring examples.
