The digits 0, 5, 7, and 9 do not appear because multiplication of two

(not necessarily different) elements of {1 2 3 4 6 8} can, with one

exception, never yield 0, 5, 7, or 9. The one exception is 3*3,
but

you'll never have two 3's in a row to yield a 9. Given the starting

configuration 12, you always append either a single even digit or an

odd digit followed by an even digit. Thus you can never have
two odd

digits in a row, so you'll never get 3*3=9.

You get arbitrarily long sequences of 8's because a sequence of three

or more 8's begets a longer sequence of 8's later on. The given

sequence shows an early subsequence of three 8's. In general,
888...

(n digits) -> 646464... (2n-2 digits) -> 242424... (4n-6 digits)
->

888... (4n-7 digits).