Let $X \sim \mathcal{N}(0, I_{n \times n})$, so that $X$ is, in distribution, $n$ independent and identically distributed draws of a $\mathcal{N}(0, 1)$ random variable. Then it is well known that
$$ \mathbf{E} \|X\|_{\ell_\infty} = \mathbf{E} \sup_{1 \leq j \leq n} |X_j|= \Theta(\sqrt{\log n}). $$ I am wondering if there are distributions on $\mathbf{R}^n$ such that $$ \mathbf{E} \|X\|_{\ell_\infty} = \Theta(1).$$
This is of course possible by taking $X$ to have support on a finite or bounded set. However I am wondering if it possible that the above holds, while $X$ is supported on all of $\mathbf{R}^n$. The coordinates need not be independent.
I'm not entirely sure what you're looking for, but here's a few different observations:
1) You probably noticed this, but this isn't possible if $X=(X_1,\ldots,X_n)$ with the $X_i$ drawn i.i.d. from any common fixed distribution (that is, independent of $i$ or $n$, which seemed to be what you wanted to avoid) with support on all of $\mathbb{R}$. This is because for each $C>0$ and all $i\in [n]$ and $n$, $\Pr(\vert X_i\vert< C)=1-\gamma_C<1$ by the assumption of full support. Then of course, \begin{equation} \mathbb{E}[\|X\|_{\ell_{\infty}}]=\mathbb{E}[\sup_{1\leq i\leq n} \vert X_i\vert]\geq C\Pr(\sup_{1\leq i\leq n} \vert X_i\vert\geq C)= C(1-(1-\gamma_C)^n). \end{equation} This last term tends to $C$ with $n$, and as $C$ is arbitrary, this implies $\mathbb{E}[\|X\|_{\ell_{\infty}}]\to \infty$ as $n\to \infty$. You could still do some sort of diagonal scaling across components to retain independence (but not identical distribution across components), but this doesn't seem too interesting to me.
2) As noted in the comments, it's not clear what exactly a satisfying example would be. One way to formalize it might be to say that all marginals are the same across components as well as $n$. This would rule out the scaling examples, for instance. Here's one potential example: to sample $X=(X_1,\ldots,X_n)$, first sample $Y$ from any distribution with full support on $\mathbb{R}$ such that $\mathbb{E}[\vert Y\vert]<\infty$, and then set $X_i=U_iY$ where the $U_i$ are i.i.d. $\mathcal{U}[-1,1]$ random variables. This will have full support on $\mathbb{R}^n$, the components have identical marginals for every $i$ and $n$, and $\vert X_i\vert\leq \vert Y\vert$ so \begin{equation} \mathbb{E}[\|X\|_{\ell_{\infty}}]\leq \mathbb{E}[\vert Y\vert]<\infty. \end{equation} In fact, offhand, I'm pretty sure you can easily show that the left side increases monotonically to $\mathbb{E}[\vert Y\vert]$.
Hope this is something like what you're looking for!