The asymptotic version of this question raised by Bjorn Poonen has been studied by [Khalid Bou-Rabee][1] for general groups, not just free groups. That is, given G a residually finite group, for each g we can ask: how large is the smallest finite group F which detects g, meaning there exists f: G -> F so that f(g) is nontrivial? Now fix a word metric on G, and ask how the maximum of this "detection number" grows as you consider words of length at most n.

See "[Quantifying residual finiteness][2]" and "[Asymptotic growth and least common multiples in groups][3]" (with Ben McReynolds) for his results. For example, as long as G is linear, the growth function is polylog, meaning asymptotically less than (log n)<sup>k</sup> for some k, if and only if G is virtually nilpotent.

To answer your question, by considering congruence quotients of SL<sub>2</sub>Z, Bou-Rabee concludes that for every word of length n in the free group F<sub>2</sub>, there is a finite group of order O(n<sup>3</sup>) where the word is not an identity.
The same bound can be obtained uniformly as follows. Ury Hadad gives a lower bound in "[On the shortest identity in finite simple groups of Lie type][5]" which implies that the shortest identity in PSL<sub>2</sub>(F<sub>q</sub>) has length at least (q-1)/3. Since the size of PSL<sub>2</sub>(F<sub>q</sub>) is order q<sup>3</sup>, this implies that every word of length at most n fails to be an identity in *one single group* PSL<sub>2</sub>(F<sub>q</sub>) of order O(n<sup>3</sup>)!

I learned this argument from Martin Kassabov and Francesco Matucci's paper "[Bounding the residual finiteness of free groups][4]". In it, they use a nice analysis of finite groups with elements of "large order" to construct a word of length n in the free group F<sub>2</sub> which is trivial in every finite group of order at most O(n<sup>2/3</sup>). This improved on the lower bound of n<sup>1/3</sup> due to Bou-Rabee and McReynolds; I believe this is now the best lower bound known.



  [1]: http://www.math.uchicago.edu/~khalid/
  [2]: http://www.math.uchicago.edu/~khalid/qrfiniteness.pdf
  [3]: http://www.math.uchicago.edu/~khalid/lcm.pdf
  [4]: http://arxiv.org/abs/0912.2368
  [5]: http://arxiv.org/abs/0808.0622