Can a language have $\Sigma^{*}$ as its syntactic monoid?
It seems to work to take $L=\{ww: w\in\Sigma^* \text{and $|ww|$ is a power of $2$}\}$.

First, we can show that if $|u|&gt;|v|$ then they are not equivalent.
Indeed, there is $x$ such that $|uxux|$ is a big power of $2$, say $2^{|u|+|v|}$.
Therefore, $uxux\in L$.

But then the length of $vxux$ cannot be power of $2$, because its difference to $2^{|u|+|v|}$ is only $|u|-|v|$. Therefore, if we take $y=xux$, we have $uy\in L$ and $vy\notin L$.

Now it suffices to show that words of the same length are not equivalent.
Let $u\neq v$ with $|u|=|v|$, and let $x$ such that $|ux|$ is a power of $2$ (and then so is $|uxux|$). Then you can see that $(u,v)\notin\sigma_L$ because $uxux\in L$ and $uxvx\notin L$.