Moving a ten-ring tower between pegs

Ten rings, numbered $1$ (smallest, on top) to $10$ (on the bottom), are stacked on one of three pegs. You may move one ring at a time and may never set a ring on top of a smaller-numbered ring. What is the least number of moves needed to rebuild the whole stack on another peg?

$1023$. This is the Tower of Hanoi. Moving $n$ rings needs moving the top $n-1$ aside, the largest once, and the $n-1$ back on top, so $M_n=2M_{n-1}+1$ with $M_1=1$, giving $M_n=2^n-1=1023$.