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 ${\overline{\vert u_{n_1}\vert\cdots \vert u_{n_t}\vert}}_{\,(b)}$ is a member of the sequence $(\vert u_n\vert)_{n\ge 0}$. 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.