The sequence begins: $2,3,6,1,8,6,8,4,8,4,8....$
(See OEIS A093095.)
$2*3=6; 3*6=1,8; 6*1=6; 1*8=8; 8*6=4,8;$ and so on.
Will there ever be a $5$? Will the sequence ever repeat?
I tried doing this by hand, and so far the only numbers I have are $1,2,3,4,6,8$: none of which can be multiplied together to get a $5$.
Which digits never occur? How does one prove this in general?
Since you are always taking the product of single digit numbers, you might want to pull out a multiplication table:
Almost the only time $m*n$ for $m,n$ single digits produces a $0, 5, 7$ or $9$ is when one (or both) of $m$ and $n$ includes a $0, 5, 7$ or $9$.
The one exception is $3*3 = 9$, but no single digit numbers multiply to give $33$.
Using this reasoning, suppose one of those four digits appeared. In fact, consider the first time one of those digits appeared. Then it must have been preceded by one of those digits, which contradicts its firstness.
Thus, $0, 5, 7$ and $9$ never appear in this sequence.