First off, we can remove the condition that $|A| \geq |F|^{\delta}$. One expects to be able to take any $\epsilon < 1$, as long as $|A| \leq |F|^{1/2}$. The most recent progress is contained in this joint work with Rudnev and Shkredov. It is known you cannot take $\epsilon =1$, say by work on the multiplication table problem or a slightly different construction in the original paper of Erdos and Szemeredi on the sum-product conjecture.

First off, we can remove the condition that $|A| \geq |F|^{\delta}$. One expects to be able to take any $\epsilon < 1$, as long as $|A| \leq |F|^{1/2}$ (not quite, see Oliver's comment below). The most recent progress is contained in this joint work with Rudnev and Shkredov. It is known you cannot take $\epsilon =1$, say by work on the multiplication table problem or a slightly different construction in the original paper of Erdos and Szemeredi on the sum-product conjecture.