Abstract: Let $M$ be the interval or the circle. For each real number $\alpha \in [2,\infty)$, write $\alpha=k+\tau$, where $k$ is the floor function of $\alpha$. I will discuss a construction of a finitely generated group of diffeomorphisms of $M$ which are $C^k$ and whose $k^{th}$ derivatives are $\tau$--H\"older continuous, but which are admit no algebraic smoothing to any higher H\"older continuity exponent. In particular, such a group cannot be realized as a group of $C^{k+1}$ diffeomorphisms of $M$. I will discuss the construction of countable simple groups with the same property, and give some applications to continuous groups of diffeomorphisms. This is joint work with Sang-hyun Kim.