Fix $a in mathbb{Z}$, $a otin {0,pm 1}$. A simple argument shows that for each $epsilon > 0$, and almost all (asymptotically 100% of) primes $p$, the multiplicative order of $a$ modulo $p$ exceeds $p^{frac12-epsilon}$. It is an open problem to show the same result with $frac12$ replaced by any larger constant. We show that if $a,b$ are multiplicatively independent, then for almost all primes $p$, one of $a,b,ab, a^2b, ab^2$ has order exceeding $p^{frac{1}{2}+frac{1}{30}}$. The same method allows one to produce, for each $epsilon > 0$, explicit finite sets $mathcal{A}$ with the property that for almost all primes $p$, some element of $mathcal{A}$ has order exceeding $p^{1-epsilon}$. Similar results hold for orders modulo general integers $n$ rather than primes $p$.
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