Fix a prime $q>N$ and find a prime $p$ that splits completely in the splitting field of $x^q-2$. Then $p \equiv 1 \pmod q$ and $2$ is a $q$-th power modulo $p$, so the order of $2$ is at most $(p-1)/q$. Using effective Chebotarev, you can even give an upper bound for $p$.

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Fix a prime $q>N$ and find a prime $p$ that splits completely in the splitting field of $x^q-2$. Then $p \equiv 1 \pmod q$ and $2$ is a $q$-th power modulo $p$, so the order of $2$ is at most $(p-1)/q$. Using effective Chebotarev, you can even give an upper bound for the smallest $p$.

Welcome to MO!