Decimal chain-addition $^*$ takes a finite initial "seed" string of decimal digits, say $x_1x_2...x_n$, and defines an infinite periodic string $x_1x_2...x_n...$ by iterating the following rewrite rule, taking the leftmost digit as output each time:
$$x_1x_2...x_n\ \ \to \ \ x_2...x_nx_{n+1}$$
where $x_{n+1}= (x_1 + x_2)\ (\mathrm{mod\ 10})$. In the form of a recursion this is
$$x_{n+k} = (x_k + x_{k+1})\ (\mathrm{mod\ 10}),\ \ k\ge 1.
$$
For example, starting with seed $528$, the procedure defines an infinite string $(5287057522...8593427693)^\omega$ that has a period of length $168$:
$$\begin{align} \color{blue}{\mathbf{5}28}&\\ \to\ \mathbf{2}8&7\tag{1}\\ \to\ \mathbf{8}&70\tag{2}\\ \to\ &\mathbf{7}05\tag{3}\\ & ...\\ & \to \mathbf{3}52\tag{167}\\ & \ \ \to \color{blue}{\mathbf{5}28}\tag{168}\\ & \ \quad ... \end{align} $$
Question: Is there a way (other than brute force computation) to determine the maximum period-length produced by seeds of a given length?
There seems to be an extensive literature on binary variants of this procedure (mostly related to pseudorandom number generators), under the names linear feedback shift register and lagged Fibonacci generator, but I don't see applications to the decimal case.
EDIT (per the comments): Among $n$-long seeds, $0^{n-1}1$ must produce a maximum-length period. Here are some results of my brute force computations, all using seed $0^{n-1}1$:
$$ \begin{array}{|c|c|c|}\text{Seed length, }n & \text{Maximum period-length, }l_{max} & \text{Prime factorization of }l_{max} \\ \hline 2 & 60 & 2^2 \cdot 3 \cdot 5\\ 3 & 168 & 2^3 \cdot 3 \cdot 7\\ 4 & 1560 & 2^3 \cdot 3 \cdot 5 \cdot 13\\ 5 & 16401 & 3 \cdot 7 \cdot 11\cdot 71\\ 6 & 196812 & 2^2 \cdot 3^2 \cdot 7 \cdot 11\cdot 71\\ 7 & 661416 & 2^3 \cdot 3 \cdot 7 \cdot 31\cdot 127\\ 8 & 15624 & 2^3 \cdot 3^2 \cdot 7 \cdot 31\\ 9 & 8894028 & 2^2 \cdot 3 \cdot 11 \cdot 13\cdot 71\cdot 73\\ 10& 1736327236 & 2^2 \cdot 7 \cdot 19\cdot 31\cdot 127\cdot 829\\ 11& 3712686852 & 2^2 \cdot 3 \cdot 7 \cdot 31\cdot 73\cdot 19531\\ 12& 203450520 & 2^3 \cdot 3 \cdot 5 \cdot 7\cdot 13\cdot 31\cdot 601\\ 13& 25732419240 & 2^3 \cdot 3 \cdot 5 \cdot 11\cdot 17\cdot 31\cdot 71\cdot 521\\ 14& 14417724597564 & 2^2 \cdot 3 \cdot 31\cdot 127\cdot 305175781\\ \end{array}$$
$^*$Decimal chain-addition was actually part of the "hollow nickel" cipher exposed in the 1950s.