Let $F_n$ be the free group on $n$ generators. I would like to know what the minimum length of a nontrivial word in the third term in the lower central series of $F_n$ is - (i.e. $[[F_n,F_n],F_n]$).
I know that since the exponent sum must be even, and length has the same parity as exponent sum, the length must be even. I know that it can not be less than 4 and with some simple guessing I can get elements of length 8. Could I ever find elements of length 6 or 4?
There are no such elements of length $4$ or $6$.
As I said in my comment, the only length $4$ word in $\gamma_2(F_n) = [F_n,F_n]$ is $aba^{-1}b^{-1}$ where $a^{\pm 1}$ and $b^{\pm 1}$ are free generators, and that does not lie in $\gamma_3(F_n)$.
By considering cyclic conjugates, we can see that the only essentially distinct words of reduced length $6$ in $\gamma_2(F_n)$ are $a^2ba^{-2}b^{-1}$, $aba^{-1}cb^{-1}c^{-1}$, and $abca^{-1}b^{-1}c^{-1}$, where $c^{\pm 1}$ is a third free generator.
The first of these does not lie in $\gamma_3(P)=1$ in the finite order $27$ quotient $\langle a,b \mid a^{9}=b^3=1, a^b=a^4 \rangle$ $P$ of $F_n$, so it does not lie in $\gamma_3(F_n)$.
For the other two words, we can consider their images in the quotient of $H = \langle a,b,c,\ldots \mid b=c \rangle$ of $F_n$, which is free of rank $n-1$, and we get the words $aba^{-1}b^{-1}$ and $ab^2a^{-1}b^{-2}$, neither of which lie in $\gamma_3(H)$.
So there are no words of length less than $8$ in $\gamma_3(F_n)$. Of course this type or argument will only work in relatively straightforward examples. I don't know how you would go about finding the minimal length of a word in $\gamma_k(F_n)$ for higher values of $k$.