Are there any manoeuvres that would be considered unrecoverable in a General Aviation / Training Aircraft?
Yes, easily. Just fly fast enough.

If you deflect a control surface fully at a speed above $v_A$, the risk of a structural failure is very real. After all, at $v_D$ only a third of the mechanically possible deflection is allowed (see §23.441 for the rudder and §23.455 for the ailerons). Even combining full deflections of more than one control surface at $v_A$ is not covered, as is a repeated deflection in sync with rigid body oscillations which can create excessive loads. For a definition of the design speeds, see §23.335.

The regulations require minimum control forces so the pilot feels when he/she overstresses the airframe, but they are still within the strength of a normal adult.

Also, if you fly too fast in gusty weather, you can overload the structure. This is rare because the 25ft/s gusts which must be tolerated up to $v_D$ are quite uncomfortable to fly through.