There are also the Burnside groups $B(m,n)$ for $n\ge 665$ odd: they are of exponential growth and have the law $x^n=1$ so that they cannot contain any free subgroup on two generators. The fact that they are not solvable follows by the theorem of Rosenblatt:
"A f.g. solvable group is of exponential growth if and only if it contains a free sub-semigroup on two generators."
You can find details on paragraphs VII.C.27/28 of Pierre de la Harpe's book "Topics in Geometric Group Theory" (Chicago Lectures in Mathematics, 2000)