There are $18$ such numbers. Here the PARI/GP - program and the output :
? q=0;for(m=10,99,n=m;x=digits(n);while(length(x)>1,n=prod(j=1,length(x),x[j]);x
=digits(n));if(isprime(x)==[1],q=q+1;print(q," ",m," ",x)))
1 12 [2]
2 13 [3]
3 15 [5]
4 17 [7]
5 21 [2]
6 26 [2]
7 31 [3]
8 34 [2]
9 35 [5]
10 37 [2]
11 43 [2]
12 51 [5]
13 53 [5]
14 57 [5]
15 62 [2]
16 71 [7]
17 73 [2]
18 75 [5]
?
You can also get this result by hand :
The final result must be one of the numbers $2,3,5,7$
So, the second last number must be one of $12,13,15,17,21,31,51,71$
From these numbers, only $12,15$ and $21$ can be represented by a product
of two one-digit numbers. The numbers $26,62,34,43,35,53,37,73$ are added
to the set.
Finally, only $35$ can be represented by a product of two one-digit numbers,
so $57$ and $75$ are added to the set. Those numbers are no more representable
in the desired way, so the set is complete.