I am thinking of Baumslag-Solitar groups of type $BS(1,m)=\langle a,b \mid bab^{-1} = a^m\rangle$ as a prototype.
We can think of them as follows: Start with an infinite cyclic group $\langle a\rangle$, choose an injective endomorphism $a\mapsto a^m$, and add a generator $b$ which acts on $\langle a\rangle$ by this endomorphism.
More generally, we can start with the trivial group, and finitely many times add a new generator which acts on the previous group by an injective endomorphism.
Let's call this "the generalized construction".
Examples of groups formed by the generalized construction: $BS(1,m)$ or $\langle a,b,c \mid bab^{-1}=a^m, cac^{-1}=a^n, cbc^{-1}=ba^k\rangle$
Does the generalized construction always yield solvable groups?
My guess: I guess the answer is "yes". My feeling is that those groups are repeated semidirect products of subgroups of $\mathbb{Q}$, and so they should be solvable.
A useful keyword is "ascending HNN-extension". An ascending extension of a $k$-step solvable group is $(k+1)$-step solvable. The result follows.