$\newcommand{\B}{\mathrm{B}}\newcommand{\U}{\mathrm{U}}$The extension of a group $G$ by $\B\U(1)$, where $\B$ denotes the classifying space specifies a 2-group, $\tilde{G}_2$ $$ 1\to\B \U(1)\to \tilde{G}_2\to G \to 1. $$ Similarly, extending by $\B^n\U(1)\cong K(\U(1),n)$, where $K(H,m)$ is an Eilenberg-MacLane space, specifies an $n$-group, $\tilde{G}_n$ $$ 1\to\B^n \U(1)\to \tilde{G}_n\to G \to 1. $$
Now, assuming $G$ is abelian (thanks to Qiaochu's comment), What kind of space would $\tilde{G}_?$ be, where $$ 1\to \B^n\U(1)\to\tilde{G}_?\to \B^m G\to 1\ ?$$ More specifically, taking $G=\U(1)\times\U(1)$, what is $$1\to \B^n\U(1)\to\tilde{G}_?\to \B^m (\U(1)\times\U(1))\to 1\ ?$$