Thompson's group F(http://en.wikipedia.org/wiki/Thompson_groups) is also an example: it isn't virtually solvable (actually, its commutator subgroup is simple) and does not contain a free subgroup, according to a theorem of Brin and Squier ("Groups of piecewise linear homeomorphisms of the real line.", Invent. Math. 79 (1985))).
You can find a survey about this group (and two cousins of his) written by Cannon, Floyd and Parry on Brin's webpage at www.math.binghamton.edu/matt/thompson/cfp.pdf http://www.math.binghamton.edu/matt/thompson/cfp.pdf