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/stable/2324098?seq=1#page_scan_tab_contents
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).