As is described in the title. Is there a finitely presented group $G$, with trivial profinite completion $\widehat{G}=0$, which is not amalgamated free product?


For example, the famous example Higman groups are all amalgamated free product.



Infinite finitely presented simple group (or more generally with trivial profinite completion) that is not amalgamated free product
I'm going to flesh out my comment above to an answer. [Thompson's groups](https://en.wikipedia.org/wiki/Thompson_groups) $T$ and $V$ are famous examples of finitely presented infinite simple groups. 


In [this](https://arxiv.org/abs/0708.1334) paper of Dan Farley, it is shown that $T$ and $V$ have Serre's property FA, which means that every time they act on a tree there is a global fixed point. (Farley says that this statement is originally due to Ken Brown.) When applied to the Bass--Serre tree of a splitting, it follows that any such splitting of $T$ or $V$ is trivial.


