Generalization of the Tower of Hanoi?

Question

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.

Answer 1

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.