I came to know from the paper Left Hopf Algebras by Green, Nichols and Taft that one may consider a Hopf algebra whose antipode satisfies only the left (resp. right) antipode condition.

To be more precise, let $\Bbbk$ be a field and $(B,\mu,\eta,\Delta,\varepsilon)$ a $\Bbbk$-bialgebra. We say that $B$ is a *left Hopf algebra* if there exists a linear endomorphism $S:B\to B$ such that
$$S(b_1)b_2=\varepsilon(b)1$$
for every $b\in B$ (i.e. $S$ is a left convolution inverse of the identity morphism).

In Section 3 of Left Hopf Algebras an "artificial" (in my opinion) example of such an object is provided. In a "recent" paper, A Left Quantum Group, Rodriguez-Romo and Taft provided a new example, which I still consider a bit "artificial". Is anybody aware of some more "concrete" or "natural" examples of this construction?

This question is strictly related to this other question on MSE. Maybe that was not the right place where to ask. The only answer it received concerns the quotient $B$ of the free non-commutative k-algebra $k\{X,Y\}$ by the ideal generated by the relation $XY-1$. This is, however, not a one-sided Hopf algebra because the condition $S(x)x=1$ together with $xy=1$ would imply $S(x)=y$ and hence would force $B$ to be a Hopf algebra ($x$ denotes the class of $X$ in the quotient and similarly for $y$).