  For any sheaf of sets $\mathcal F$ on $Sm/k$, it is well known that the
universal $\mathbb A^1$-invariant quotient of $\mathcal F$ is given as the
colimit of sheaves $\mathcal S^n(\mathcal F)$ where $\mathcal S(F)$ is the
sheaf of naive $\mathbb A^1$-connected components of $\mathcal F$. We show that
these infinite iterations of naive $\mathbb A^1$-connected components in the
construction of universal $\mathbb A^1$-invariant quotient for a scheme are
certainly required. For every $n$, we construct an $\mathbb A^1$-connected
variety $X_n$ such that $\mathcal S^n(X_n)\neq \mathcal S^{n+1}(X_n)$ and
$\mathcal S^{n+2}(X_n)=*$.
