An example of an infinite non-abelian solvable group This isn't exactly a homework problem-- it's on a sample exam.  
My first instinct is to look to matrix groups, since they are very often non-abelian and infinite, but I haven't had any luck. 
 A: I confess that the only example that comes to mind is the one Joel mentions in the comments: the subgroup $B$ of $GL_n(K)$, where $K$ is an infinite field and $n \geq 2$, consisting of invertible upper triangular matrices. Let's work this out when $n = 2$. Then
\[
B = \left\{\begin{pmatrix}
a & b \\
0 & d
\end{pmatrix} \Bigg|\  a, d \in K^*, b \in K\right\}.
\]
This is infinite because $K$ is, and if $d$ is an element of $K^*$ not equal to $1$ then
\[
\begin{pmatrix}
1 & 0 \\
0 & d
\end{pmatrix} \qquad \text{and} \qquad
\begin{pmatrix}
1 & 1 \\
0 & 1
\end{pmatrix}
\]
do not commute. There's a homomorphism $B \to K^* \times K^*$ sending a general element as above to the diagonal $(a, d)$. It's surjective and the kernel is the normal subgroup
\[
U = \left\{\begin{pmatrix}
1 & b \\
0 & 1
\end{pmatrix} \Bigg|\   b \in K\right\}
\]
of $B$. And as yoyo says in the comments, $U$ is isomorphic to the additive group of $K$. We get an abelian tower
\[
B \supset U\supset \{I\}
\]
If $n > 2$ then there are more steps in the tower; I think this example is written out in the general case in first chapter of Lang's Algebra. For $n = 3$ our $U$ is the Heisenberg group, which is interesting enough.
A: How about $S_3 \times \mathbb Z$.
