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 I wouldn't be surprised, ifon GRH Hooley proved that for each 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.
AssumingFor 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 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 an elementarya 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 ChebatorevChebotarev).
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.