The more common name for a Saturday number is
a Polydivisible number
There is a good argument for believing that they cannot grow to any length:
If you remove the last digit from a Polydivisible number you get a smaller Polydivisible number. Conversely, you can only get a length $n$ Polydivisible number if you append a digit $d$ to a length $n-1$ Polydivisible number $q$. For that to work you need $q,d$ to be such that $10q+d\equiv 0 \bmod n$. Only $10$ values are possible for the digit $d$, so no more than $10$ out of $n$ residue classes for $10q \bmod n$ allow this extension to work. Assuming that each residue class is equally probable, the probability that you can extend a Polydivisible number is $10/n$, so when $n>10$ you expected there to be fewer Polydivisible numbers remaining each time you try to extend them.
The longest Saturday number is:
3608528850368400786036725, which is 25 digits long.
To be honest, I wrote a computer program to find it, and only found the common name for this type of number after I googled the number that I found. These numbers are listed in the OEIS.
For those interested, here is the simple C# program I wrote:
public static void Main()
List<BigInteger> current = new List<BigInteger>();
List<BigInteger> next = new List<BigInteger>();
for (int i = 1; i <= 9; i++)
int length = 1;
while(current.Count > 0)
foreach (var n in current)
for( int d=0; d<=9; d++)
var n2 = n * 10 + d;
if(n2 % length == 0)
var t = current;
current = next;
next = t;