A very simple and elementary way to define a total ( bi-invariant ) ordering on the free group is to use the Magnus transformation which, an alphabet $X=\{x_1,\cdots ,x_n\}$ being given, sends a reduced word $g=x_{i_1}^{\epsilon_1}x_{i_2}^{\epsilon_2}\cdots x_{i_k}^{\epsilon_k} \in \Gamma(X)$ ( with $\epsilon_i\in \{-1,1\}$ ) to the series $$ S=S(x_{i_1}^{\epsilon_1}x_{i_2}^{\epsilon_2}\cdots x_{i_k}^{\epsilon_k})=(1+x_{i_1})^{\epsilon_1}(1+x_{i_2})^{\epsilon_2}\cdots (1+x_{i_k})^{\epsilon_k}\in \mathbb{Z}<<X>> $$ ( it can be shown to be injective ). Then ( if $x_{i_1}^{\epsilon_1}x_{i_2}^{\epsilon_2}\cdots x_{i_k}^{\epsilon_k}\not=1$ ), develop it as $$ (1+x_{i_1})^{\epsilon_1}(1+x_{i_2})^{\epsilon_2}\cdots (1+x_{i_k})^{\epsilon_k}=1+\cdots + \mathbf{\lambda_w\,w} +\cdots $$ where $w\in X^{*}$ is the smallest word ( for some lexicographic ordering given by a total ordering e.g. by $x_1<\cdots <x_n$ on $X$ ) among words of smallest length in the support of $S$.
Then, if $\lambda_w\in \mathbb{N}_{\geq 1}$, say that $g$ is positive. In this way, one defines a "strictly positive cone" $P\subset \Gamma(X)$ and checks that it satisfies the three conditions ( for all $g\in \Gamma(X)$ ) $$ P.P\subset P\ ;\ gPg^{-1}\subset P \ ;\ P\cap P^{-1}=\emptyset\ ;\ P\cup P^{-1}=\Gamma(X) \setminus \{1\} $$ hence defining a strict total ordering on $\Gamma(X)$. This construction works mutatis mutandis when $X$ is an infinite alphabet.

As the OP was aking for examples, here is a very simple and elementary way to define a total ( bi-invariant ) ordering on the free group using the Magnus transformation which, an alphabet $X=\{x_1,\cdots ,x_n\}$ being given, sends a reduced ( and, in fact, any ) word $g=x_{i_1}^{\epsilon_1}x_{i_2}^{\epsilon_2}\cdots x_{i_k}^{\epsilon_k} \in \Gamma(X)$ ( with $\epsilon_i\in \{-1,1\}$ ) to the series $$ S=S(x_{i_1}^{\epsilon_1}x_{i_2}^{\epsilon_2}\cdots x_{i_k}^{\epsilon_k})=(1+x_{i_1})^{\epsilon_1}(1+x_{i_2})^{\epsilon_2}\cdots (1+x_{i_k})^{\epsilon_k}\in \mathbb{Z}<<X>> $$ ( it can be shown to be injective ). Then ( if $x_{i_1}^{\epsilon_1}x_{i_2}^{\epsilon_2}\cdots x_{i_k}^{\epsilon_k}\not=1$ ), develop it as $$ (1+x_{i_1})^{\epsilon_1}(1+x_{i_2})^{\epsilon_2}\cdots (1+x_{i_k})^{\epsilon_k}=1+\cdots + \mathbf{\lambda_w\,w} +\cdots $$ where $w\in X^{*}$ is the smallest word ( for some lexicographic ordering given by a total ordering e.g. by $x_1<\cdots <x_n$ on $X$ ) among words of smallest length in the support of $S$.
Then, if $\lambda_w\in \mathbb{N}_{\geq 1}$, say that $g$ is positive. In this way, one defines a "strictly positive cone" $P\subset \Gamma(X)$ and checks that it satisfies the three conditions ( for all $g\in \Gamma(X)$ ) $$ P.P\subset P\ ;\ gPg^{-1}\subset P \ ;\ P\cap P^{-1}=\emptyset\ ;\ P\cup P^{-1}=\Gamma(X) \setminus \{1\} $$ hence defining a strict total ordering on $\Gamma(X)$. This construction works mutatis mutandis when $X$ is an infinite alphabet.