I am reading a paper where the author say "let $n\geq2$ and $\ell_\infty^n:=\Bbb R^n$ with the sup norm, like usual". Why is that notation usual?
I was asking that because, for me, $\ell_\infty=\{(x_j)\in\Bbb R^\Bbb N~|~\sup|x_j|<\infty\}$ and $\ell_\infty^n=\ell_\infty\times\cdots\times\ell_\infty$ $n$-times. So the dimension of $\ell_\infty^n$ would be $\infty$.
Superscripts need not mean powers. The sphere $S^n$ is not the $n$th power of anything. Neither is $L^2$ the square of $L^1$. In papers on dynamical systems, $f^n$ is more likely to be the $n$th iterate $f\circ \cdots\circ f$ than the $n$th power of $f$. Notation does not rule over us; we use whatever notation is convenient in a given context.
If someone works with finite-dimensional normed spaces a lot, they will use $\ell_p^n$ because it's a short and convenient notation. The superscript traditionally denotes the dimension of a manifold or a vector space, and we still have to put $p$ somewhere to indicate what norm it has, so it goes in the subscript. (And $\ell$ is probably by similarity with the function space $L_p$, where it comes from "Lebesgue integrable".)
On the other hand, I have little use for the $n$th Cartesian power of the space $\ell_\infty$ (it's trivially isometric to $\ell_\infty$). If I ever needed it, I'd write $(\ell_\infty)^n$.