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 --
Moreover I wouldn't be surprised, if for each fixed $L$ there is a positive density of primes, where the order of 2 mod p, $\text{ord}_p(2)$ is $\frac{p-1}{L}$.
Assuming GRH, 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/openurl?issn=0002-9890&volume=96&issue=4&spage=361&date=19$
In short, the published solution gives an elementary proof based on Dirichlet's theorem on primes in progression, and mentions that other proofs exist that are "not completely elementary", (e.g. including Chebatorev).