Two properties and you are done

Consider $s(n)$ and $s(2n)$ . Each digit $k$ of $n$ contributes at most $2k$ to $s(2n)$ . Tthis contribution is reduced if $2k \ge 10$ , so that carrying is required. Thus $s(2n) \le 2s(n)$ .

Now consider $s(n)$ and $s(11n)$ . Each digit $k$ of $n$ contributes at most $2k$ to $s(11n)$ , since it contributes $k$ as part of $n$ , and another $k$ as part of $10n$ . This contribution is reduced if carrying is required in the columns involving $k$ when $n$ and $10n$ are added together. Thus $s(11n) \le 2s(n)$ .

Hence $s(44n) \le 2s(4n) \le 4s(2n) \le 8s(n)$ . If we are to have $s(44n) = 8s(n)$ , as in this case, we must have no carrying when doubling $n$ to $2n$ , when doubling $2n$ to $4n$ and when multiplying $4n$ by $11$ . For the first of these conditions to happen, all the digits of $n$ must be taken from $0,1,2,3,4$ . For the first two of these conditions to occur, all the digits of $n$ must be taken from $0,1,2$ , so that all the digits of $4n$ will be one of $0,4,8$ . Thus all three conditions will be true if all the digits of $n$ are either $1$ or $0$ , or else if no digit $2$ is adjacent to any digit except $0$ .

Whatever the value of $n$ , since it only contains the digits $0,1,2$ , there is no carrying when calculating $3n$ . Thus $s(3n) = 3s(n) = \boxed{300}$ . This value can be achieved, since the value of $n$ written as $100$ $1$ s in a row is an example of $n$ where $s(n) = 100$ .

We will use two important properties:

For $N = \overline { { a }_{ n }{ a }_{ n-1 }...{ a }_{ 0 } }$ , we have

$S({ N }_{ 1 } + { N }_{ 2 }) \le { S(N }_{ 1 }) + S({ N }_{ 2 }) \\ S({ N }_{ 1 }{ N }_{ 2 }) \le S({ N }_{ 1 })S({ N }_{ 2 })$ .

By using above two properties and $s(n) = 100, s(4n) = 800$ ; we have

$S\left( 3n \right) \le 3S\left( n \right) = 300$

and

$\large\ 800 = S\left( 11.3n + 11n \right) \le S\left( 11.3n \right) + S\left( 11n \right) \le S\left( 11 \right) S\left( 3n \right) + S\left( 11 \right) S\left( n \right) = 2S\left( 3n \right) + 200. \\ \Rightarrow S\left( 3n \right) \ge 300.$

Thus, $\large\ S(3n) = 300$ .