Let $\Omega \subseteq \mathbb{R}^N$ be a bounded open set with $C^1$ boundary (I'll call this kind of sets regular domains). I know that given $u\in W^{1,p}(\Omega)$ the following inequality (Gagliardo-Nirerberg-Sobolev) holds for any $r<N$ (and also for $r=N=1$):
$$||u||_{L^{r^*}(\Omega)}\leq C ||u||_{W^{1,r}(\Omega)}, \ \ \ \ r^*:=\frac{Nr}{N-r}.$$
This implies pretty easily that if $u\in W^{1,p}(\Omega)$ with $p\geq N$ then $u\in L^q(\Omega)$ for any $q\in [1,\infty)$.
What about $q=\infty$?
If $p=N\neq 1$ I know that a valid counterexample is $\log(\log(1+1/|x|))$.
If $p=N=1$ then $W^{1,1}(a,b)=AC(a,b)$ (absolutely continous functions) so no counterexample is possible.
If $p>N$ then I don't know if a counterexample exists and I need some help.
You can embed $W^{1,p}(\Omega)$ into $C^{0,\alpha}(\Omega)$ for $p> N$ and some $\alpha\in(0,1)$ (This is Morrey's inequality). Therefore, we get that for every $u\in W^{1,p}(\Omega)$, there is a bounded (Because hölder-continuous on bounded set) representation which is a.e. equal to $u$.