Do functions with slower growth than inverse Ackermann appear in runtime bounds? Some complicated algorithms (union-find) have the nearly-constant inverse Ackermann function that appears in the asymptotic time complexity, and are worst-case time optimal if the nearly constant inverse Ackermann term is ignored. 

Are there any examples of known algorithms with running times that involve functions that grow fundamentally slower than inverse Ackermann (e.g. inverses of functions that are not equivalent to Ackermann under polynomial or exponential etc. transformations), that give the best-known worst-case time complexity for solving the underlying problem?
Seth Pettie came up with an algorithm for computing the sensitivity of a minimum spanning tree in time $O(m\log \alpha(m,n))$, improving on an algorithm of Tarjan which computes the same in time $O(m\alpha(m,n))$. (Compare this to Chazelle's $O(m\alpha(m,n))$ algorithm for computing the minimum spanning tree itself.) The sensitivity problem asks to compute, for a given graph and a given minimum spanning tree, by how much each edge weight can change without changing the minimum spanning tree.

(Thanks to Tsvi Kopelowitz for this reference.)