Let $G$ be a compact connected Lie group and let $P$ be a principal $G$-bundle over $K$. The gauge group of $P$ is the topological group of automorphisms of $P$. Consider all principal $G$-bundles $P$ over $K$. It is known that if $K$ is a finite complex and $n<\infty$, the number of $A_n$-types of gauge groups of $P$ is finite even when the number of bundles is infinite \cite{CS,Ts}. We prove that this finiteness typically breaks by proving that the number of $A_\infty$-types of gauge groups of $P$ is infinite if $G$ is simple, $K=S^n$ and $|\pi_{n-1}(G)|=\infty$.