Erich Friedman's two parter:

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).