Let $$\epsilon_m=\max \sum \frac{1}{n_i}$$ where the maximum is taken over all finite sequences $m<n_1<\cdots<n_k$ for which there exist congruences $a_i\pmod{n_i}$ such that no integer satisfies two such congruences.

The work of de la Bretèche, Ford, and Vandehey [BFV13] implies $$L(m)^{-1+o(1)}< \epsilon_m < L(m)^{-\sqrt{3}/2+o(1)},$$ where $L(m)=\exp(\sqrt{\log m\log\log m})$. The lower bound is implicit in their construction.