It is clear that cyclic groups have the property that they cannot be written in a non-trivial way as an amalgamated free product or as an HNN-extension.
Can someone please provide us examples of torsion-free 2-generated groups having this property? Any comments related to this question are welcome!
[I'm aware of Serre's Theorem 15 in Trees.]
Edit: The term "in a non-trivial way" means the following. For an amalgamated free product $G=H*_C K$ it is required $H\ne C\ne K$. For an HNN-extension $G=HNN(H,A,B,t)$ it is required $A\ne H\ne B$.