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Given positive integers $A_1,\ldots,A_t$ and $b\ge 2$,
we write $\overline{A_1\cdots A_t}_{(b)}$ for the integer
whose base-$b$ representation is the concatenation
of the base-$b$ representations of $A_1,\ldots,A_t$.
In this paper, we prove that if $(u_n)_{n\ge 0}$ is
a binary recurrent sequence of integers satisfying some mild
hypotheses, then for every fixed integer $t\ge 1$, 
there are at most finitely many nonnegative integers $n_1,\ldots,n_t$ 
such that ${\overline{|u_{n_1}|\cdots |u_{n_t}|}}_{\,(b)}$ is
a member of the sequence $(|u_n|)_{n\ge 0}$.  In particular,
we compute all such instances in the special case that $b=10$, $t=2$, 
and $u_n=F_n$ is the sequence of Fibonacci numbers. 

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