Thompson's group $F$ is totally ordered.  See [here][1] for a description of all possible bi-orderings.  Indeed, all diagram groups are totally orderable (see [here][2]).  The pure braided Thompson group $BF$ also bi-orderable (see [here][3]).


  [1]: http://arxiv.org/pdf/0808.1688.pdf
  [2]: http://arxiv.org/pdf/math/0305153v1.pdf
  [3]: http://www-ma4.upc.edu/~burillo/Papers/biorder.pdf