Sobolev Embedding (Case: p=N)

First of all, the one-dimensional example does not work. The function $\log\log (1+1/t)$ does not belong to $W^{1,1}((0,1))$. Indeed, the space $W^{1,1}((0,1))$ consists of the antiderivatives of Lebesgue integrable functions on $(0,1)$, and all such functions are bounded. One can also give an explicit estimate of Poincaré type, $$ \sup_{(0,1)} \left|f - \int_0^1 f\right| \le \int_0^1 |f'| $$

But in dimensions $N\ge 2$ the function $u(x)=\log\log(1+|x|^{-1})$ is indeed the standard example used to show that $W^{1,N}$ does not embed into $L^\infty$.

The question was whether adding the assumption $u\in L^\infty$ could save the embedding. It does not, as can be seen by considering $\min(u,M)$ for arbitrarily large $M$.

There's a more general reason for why the assumption of boundedness could not help here. The space $C^\infty_c$ is dense in $W^{1,N}$, as in any other Sobolev space with $p< \infty$. If an estimate of the form $F(u)\le C\|u\|_{X}$ holds on a dense subset of a normed space $X$, then it holds for all $u\in X$.