Persistent Number

Given a number $x$, we define a function $p(x)$ as the product of the digits of $x$. We can then form a sequence x, p(x), p(p(x))... . The persistence of $x$ is then defined as the index ($0$-based) of the first single digit number in the sequence. For example, using $99$, we get the sequence 99, 9 \cdot 9 = 81, 8 \cdot 1 = 8. Thus, the persistence of $99$ is $2$. You will be given $n$, and you must find its persistence.