Here are a few aspects that have not yet been mentioned by
Felipe, Joe and Vesselin.

a) Assuming GRH, 2 is a primitive root mod $p$ for a positive
density of primes. (The density is 0.373955, known as Artin's constant).
For a great survey on this see Pieter Moree,
Integers, Volume 12A (2012), A13: Artin's Primitive Root Conjecture -- A
Survey --
http://www.integers-ejcnt.org/vol12a.html

Moreover, on GRH Hooley proved
that for fixed $L$ there is a
positive density of primes, where the order of 2 mod p, $\text{ord}_p(2)$ is $\frac{p-1}{L}$. ($L$ is also called the index).
Murata worked out that this density roughly decreases as $1/L^2$.
The sum (over $L$) of these densities converges to 1.

For a convenient statement of this see Pappalardi,
On Hooley's Theorem with weights. Rend. Sem. Mat. Polit. Univ. Torino 53 No. 4 (1995) 375-388.
http://www.mat.uniroma3.it/users/pappa/papers/Rend_Sem_Torino_53_1995.pdf

For another statement, assuming GRH, that the average value of the order is quite large, see for example Thm. 2 of Kurlberg and Pomerance:
ON A PROBLEM OF ARNOLD: THE AVERAGE
MULTIPLICATIVE ORDER OF A GIVEN INTEGER

https://math.dartmouth.edu/~carlp/arnoldfinal.pdf

b) Your question whether $\frac{p-1}{\text{ord}_p(2)}$ can actually be
bounded has appeared in the Monthly a while ago:

E3216
Jon Froemke, Jerrold W. Grossman, O. P. Lossers
The American Mathematical Monthly, Vol. 96, No. 4 (Apr., 1989), p. 361.

http://www.jstor.org/stable/2324098?seq=1#page_scan_tab_contents

In short, the published solution gives a proof that it is not bounded, based on
Dirichlet's theorem on primes in progression, and mentions that other
proofs exist that are "not completely elementary",
(e.g. including Chebotarev).

c) Unconditionally, one can show that $\text{ord}_p(2)\geq p^{1/2 + \varepsilon(p)}$, for most primes, where $\varepsilon(p)$ tends to 0, as $p$ tends to infinity. See Erdos and Ram Murty,
On the order of a mod p, CRM Proceedings and Lecture Notes, Volume 19, (1999) pp. 87-97.
http://www.mast.queensu.ca/~murty/erdos.dvi

Moreover, one can show that for a positive proportion of primes
the order is a bit larger, $> p^{0.677}$, where the exponent comes from a result of Baker and Harman on the largest prime factor of $p-1$.
See Lemma 20, Kurlberg and Pomerance, On the period of the linear congruential and power generators, P. Kurlberg and C. Pomerance, Acta Arith. 119 (2005), 149–169.
http://www.math.dartmouth.edu/~carlp/PDF/par13.pdf

In other words, what is known unconditionally is much weaker than what is known on GRH.