Let $\Omega$ be a bounded domain in $\mathbb R^2$. By the Sobolev embedding theorem, if $k>\frac np$ (in this case $k>\frac 2p$) then

$u\in W^{k,p}(U) \implies u\in C^{k-[\frac 2 p]-1,\gamma}(U)$

for a certain $\gamma$; where $[\frac 2 p]$ is the integer part of $\frac 2p$.

If $k=1$ and $p=2$ then a function $u\in W^{1,2}(U)$ is not necessarily in $L^\infty(U)$, like the function $\log|\log|x^2+y^2||$ shows. (Let $U$ be the disk centered at the origin of radius $\frac 12$.)

What is an example that shows that if $k=2$ and $p=2$ then for a function $u\in W^{2,2}(U)$, $|\nabla u|$ is not necessarily in $L^\infty(U)$?

More generally, is there a standard way to construct such example from the previous one, that is $\log|\log|x^2+y^2||$?