Fewest moves to shift a 64-disc Hanoi tower

Sixty-four discs of increasing size are stacked on one of three pegs, largest at the bottom. A move takes the top disc of one peg to another peg, and a disc may never rest on a smaller one. What is the least number of moves needed to transfer the whole stack to another peg?

$2^{64}-1$ (about $1.8\times10^{19}$). To move $n$ discs you must first park the top $n-1$ elsewhere, move the largest once, then restack the $n-1$ on top, so $M_n=2M_{n-1}+1$ with $M_1=1$, which solves to $M_n=2^n-1$.