What is an example of a Hopf algebra $(H,\Delta,\epsilon)$ containing an invertible element $h$ which is not grouplike: An element $h \in H$ such that
$$
\Delta(h) \neq h \otimes h\qquad\text{(not grouplike)}
$$
and such that there exists a $h^{-1} \in H$ with $hh^{-1} = h^{-1}h = 1$ (invertible).



Hopf algebra with a non-grouplike invertible element
Let $L$ be a finite-dimensional $p$-nilpotent restricted Lie algebra over a field of characteristic $p>0$ and consider its restricted enveloping algebra $u(L)$. Then the only group-like element of the Hopf algebra $u(L)$ is 1. On the other hand, every element of $u(L)$ that is not in the kernel of the counit is invertible.


