For $p$ prime denote by $\mathsf{ord}_p(2)$ the multiplicative order of $2$ modulo $p$.

Does there exist $N > 0$ such that, for ALL primes $p$, $\mathsf{ord}_p(2)$ is at least $\frac{(p-1)}{N}$?

If there are infinitely many Mersenne primes then the answer is "no". Since the order of $2$ modulo a Mersenne prime $p=2^k-1$ is only $k$, which is not greater than $\frac pN$ for sufficiently large $p$.

Is there a proof that the answer is "no"?

It is possible that the following question answers this, but it wasn't clear to me: The critical exponent in the multiplicative order of 2 modulo primes

Thanks Steven Galbraith