ELMO SL 2018 C2

We say that a positive integer $n$ is $m$ -expressible if it is possible to get $n$ from some $m$ digits and the six operations $+,-,\times,\div$ , exponentiation $^\wedge$ , and concatenation $\oplus$ . For example, $5625$ is $3$ -expressible (in two ways): both $5\oplus (5^\wedge 4)$ and $(7\oplus 5)^\wedge 2$ yield $5625$ .

Does there exist a positive integer $N$ such that all positive integers with $N$ digits are $(N-1)$ -expressible?

Proposed by Krit Boonsiriseth