Going a bit outside the box here.
If we define a "Fibonacci-base" representation as
$$({\ldots}a_2a_1a_0)_F=\sum_{k=1}^\infty a_kF_k$$
where $a_k\in\{0,1\}$ and $F_k$ are the Fibonacci numbers, then applying the Shanks transformation to ${\ldots}111_F$ gives $-1$. Applying the Shanks transformation to the ordinary binary representation ${\ldots}111_2$ also gives $-1$.