Generalization of the Tower of Hanoi? What is the least possible number of steps for the Tower of Hanoi with $n$ discs and an arbitrary number $k$ of towers?
For example, Tower of Hanoi with $4$ towers, $5$ towers, etc.
 A: Interestingly enough, it seems that we don't actually know! 
From Wikipedia:

Although the three-peg version has a simple recursive solution as
  outlined above, the optimal solution for the Tower of Hanoi problem
  with four pegs (called Reve's puzzle), let alone more pegs, is still
  an open problem.

There is, however, a presumed optimal solution for the puzzle with Four Spindles, also to be found in the same Wikipedia article.
A: The article by Demontis is totally flawed, for more details see https://doi.org/10.1142/S1793830919500496. In 2014 Thierry Bousch published a solution to The Reve's puzzle (four pegs version), see https://www.math.u-psud.fr/~bousch/preprints/revespzl.pdf. The FS conjecture for Tower of Hanoi with five or more pegs remains an open problem.
A: The presumed optimal solution for four or more pegs, predicted by the Frame-Stewart algorithm in 1941, has been recently proved optimal after about 80 years by Roberto Demontis in this 2019 paper What is the least number of moves needed to solve the k-peg Towers of Hanoi problem? (you can find a free PDF here).
