The fundamental group of an open set in $\mathbb{R} ^n$ does not have nilpotent elements.

Question

I am studying a little of basic algebraic topology and I thought that this statement could be true. If you have an open connected set $U \subset \mathbb{R}^m$ and a loop $\gamma$ that is not null-homotopic (homotopic to a constant) then $\gamma \ast \gamma ... \ast \gamma$ (n-times) is never null-homotopic for $n \in \mathbb{N}$.