HNN extensions which are free products which HNN-extensions are free products? this question is related with another still unsolved about Nielsen-Thruston-reducibility and connected-sum-irreducibility of 3d-torus- bundles... 
 A: This might help.
Lemma If $A$ does not split freely and $C$ is a non-trivial subgroup of $A$ then the HNN extension $G=A*_C$ does not split freely.
The proof uses Bass--Serre theory---see Serre's book Trees from 1980.
Proof.  Let $T$ be the Bass--Serre tree of a free splitting of $G$.  Because $A$ does not split freely, $A$ stabilizes some unique vertex $v$.  But $C$ is non-trivial, so $C$ also stabilizes a unique vertex, which must be $v$.   Therefore,  $G$ stabilizes $v$, which means the free splitting was trivial. QED
A similar argument shows the following.
Lemma If $ A*_C $ splits non-trivially as an amalgamated free product $ A' *_{C'} B'$ then either $A$ splits over $C'$ or $C$ is conjugate into $C'$.
A: Let me add an explicit partial solution to the question above: for a torus bundle $E$ over the circle, the fundamental group of $E$ can't be a free product of groups, because if it were, the fundamental subgroup of the fiber would be a free product (by the Kurosch's theorem) which is impossible for the torus, then $E$ isn't a connected sum, hence irreducible.
At least the HNN extensions which are free products can't be torus bundles, and in fact, no other surface bundles unless the surface be the 2-sphere  
