For p prime denote by ord_p(2) the multiplicative order of 2 modulo p.

Does there exist N > 0 such that, for ALL primes p, ord_p(2) is at least (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 p/N 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