[seqfan] Multiply my digits

[Éric Angelini]

[this msg was X-posted to Math-Fun
a few hours ago]
Hello SeqFans
Say a(n) is the product of the digits of a(n+1); and this a(n+1) is the product of the digits of a(n+2); and this a(n+2) etc. 
I could not go further than a(n+3) with 8, 42, 1372, 22777,... 
 Of course a way to expand this seq. would be to test all integers written with the digits 1,2,2,7,7,7 or with the digits 1,1,2,2,7,7,7 or with the digits 1,1,1,2,2,7,7,7 etc. (as their product would still be the same = 1372), then check if one of those integers might be the product of single digits only. I got stuck as this task is quickly impossible to execute on a smart- phone! 
Perhaps starting with another a(1) than 8 is more promising? And if there are many infinite such seqs, which one would be the lexico- graphically earliest? [pardon me of this is old hat] 
Best, 
É.