To supplement David's answer, the 2nd option, I can add the following heuristics. If we fix a positive integer $n$ and take irreducible polynomials $$ f_k(x) =A\frac{x^{k+1}-1}{x-1}+B =A(x^k+x^{k-1}+\dots+x)+B, \qquad k=1,\dots,n, $$ subject to the condition that $f_1(x)\dots f_n(x)/m$ is not an integer-valued polynomial for all $m>1$ (take, for example, $A=3$ and $B=1$), then Schinzel's Hypothesis H implies that the numbers $f_1(R),\dots,f_n(R)$ are simultaneously prime for infinitely many positive $R\in\mathbb Z$. There is no guarantee however that $R$ assumes the form $10^k$. I owe this idea to Frictionless Jellyfish who removed his/her related comment to this post (which I would definitely accept as answer).

To supplement David's answer, the 2nd option, I can add the following heuristics. If we fix a positive integer $n$ and take irreducible polynomials $$ f_k(x) =A\frac{x^{k+1}-1}{x-1}+B =A(x^k+x^{k-1}+\dots+x)+B, \qquad k=1,\dots,n, $$ subject to the condition that $f_1(x)\dots f_n(x)/m$ is not an integer-valued polynomial for all $m>1$ (take, for example, $A=3$ and $B=1$), then Schinzel's Hypothesis H implies that the numbers $f_1(R),\dots,f_n(R)$ are simultaneously prime for infinitely many positive $R\in\mathbb Z$. There is no guarantee however that $R$ assumes the form $10^k$. I owe this idea to Frictionless Jellyfish who removed his/her related comment to this post (which I would definitely accept as answer).