Minimum number of operations (divide by 2/3 or subtract 1) to reduce $n$ to $1$
Just a bit of statistics: Up to a billion, worst case is 644,972,543 with 44 moves. Up to 4 billion, worst case is 3,386,105,855 with 48 moves. 99% of all numbers to 36 moves or fewer, 99.99% take 39 moves or fewer.
The simple algorithm "Divide by 3 if possible, else divide by 2 if possible, else subtract 1" has the worst case numbers 3 * 2^k - 2 taking 2k steps and 3 * 2^k - 1 taking 2k + 1 steps, which is substantially worse.