The more common name for a Saturday number is
>! a [Polydivisible number][1]

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][2].

For those interested, here is the simple C# program I wrote:

      using System;
      using System.Collections.Generic;
      using System.Numerics;

      namespace TempProg
      {
         class PSEsaturday
         {
            public static void Main()
            {
               List<BigInteger> current = new List<BigInteger>();
               List<BigInteger> next = new List<BigInteger>();

               for (int i = 1; i <= 9; i++)
                  current.Add(new BigInteger(i));

               int length = 1;
               while(current.Count > 0)
               {
                  length++;
                  foreach (var n in current)
                  {
                     for( int d=0; d<=9; d++)
                     {
                        var n2 = n * 10 + d;
                        if(n2 % length == 0)
                        {
                           next.Add(n2);
                           Console.WriteLine(n2);
                        }
                     }
                  }
                  var t = current;
                  current = next;
                  next = t;
                  next.Clear();
               }
            }
         }
      }




  [1]: https://en.wikipedia.org/wiki/Polydivisible_number
  [2]: https://oeis.org/A109032