Is there a "most unstable rigid body shape" when spun about its intermediate axis? Is there a shape of a solid, that provides the quickest and most violent deviation of its axis of rotation, when spun around its intermediate axis? What would be this shape and does it depend on a particular length ratio between the three axis, in this case is there a sweet spot ratio for getting the most unstable behaviour?
 A: The rotation property of the solid only depends on its principal moment of inertia: $I_1\geq I_2\geq I_3$, so you can always revert to the case where it’s a brick, an ellipsoid or pretty much any shape you want. You’ll get the corresponding principle moments of inertia by squeezing it appropriately about the principle directions.
I would understand “the most unstable” by maximizing the Liapunov coefficient $\lambda$ which is the inverse timescale at which the rotational axis diverges from the intermediate axis. It is obtained by linearizing Euler’s equation about the unstable equilibrium. You get:
$$
\lambda = \sqrt{\frac{(I_1-I_2)(I_2-I_3)}{I_1I_3}}\Omega
$$
with $\Omega$ is angular velocity which is directed along the unstable intermediate axis. The fact that it is proportional to $\Omega$ is not surprising as by increasing the initial angular velocity necessary decreases the timescale from dimensional analysis (only timescale of the problem).
The answer to you question thus lies in the geometrical prefactor in front:
$$
C = \sqrt{\frac{(I_1-I_2)(I_2-I_3)}{I_1I_3}}
$$
which you must seek to maximize by squeezing appropriately your shape. Note that it can take any value you want, especially arbitrarily large ones. The problem would be more interesting if you were to add constraints like fixing volume, mass, minimum/maximum radius etc.
Hope this helps.
