Suppose $d_1, d_2$ are two fixed coprime integers, $\frac{d_1}{d_2} \neq \pm 1$. Given any $n > 0$, can we find a prime number $p$ such that the order of $d_1d^{1}_2$ in the multiplicative group of the field $\mathbb{Z}/p\mathbb{Z}$ be greater than $n$?

The answer to your question is "yes" (cf. Douglas Zare's comment). In fact, for all sufficiently large primes $p$, the order of $d_1 d_2^{1}$ is greater than $n$. Here, "sufficiently large" means greater than $d_1^n$ and $d_2^n$. 

