Given a topological group $G$ we can define $G_1$ to be the subgroup given by the intersection of all clopen sets containing the neutral element. For $n>1$ we define recursively $G_n:=(G_{n-1})_1$.
Now I'm looking for a group $G$ such that $G_n \ne G_{n+1}$ for all $n$. Is there an easy example?
By the way: We have $G_n = G_{n+1}$ iff $G_n$ is the connected component of the neutral element in $G$.