Completely forbidden transitions $\newcommand{\bra}[1] {\left< #1 \right|}
 \newcommand{\ket}[1] {\left| #1 \right>}
$In molecular physics and spectroscopy physicists often calculate the dipole transition elements between two particular bound states. Usually this is done via the first order perturbation to the Hamiltonian leading to a transition amplitude of the form $\bra{i}\vec{E}\cdot\vec{r}\ket{f}$. If the state $\bra{i}$ and the state $\ket{f}$ have definite parities this can lead to "forbidden" transitions.
I am curious as to the mathematical conditions that the state $\bra{i}$ and the state $\ket{f}$ have to satisfy in order for the transition to be forbidden to all orders. Is there a possible symmetry argument that would lead to transitions forbidden to all orders? 
Are there any measured / hypothesised transitions that are not allowed to all orders?
If one takes QED into account allowing for virtual states between eigenstates (for example the two photon $2s \to 1s$ transition in hydrogen) - is it possible to have a fully forbidden transition?
 A: There are two ways to have absolutely forbidden transitions, one rigorous, the other only valid in the thermodynamic limit.
One class of forbidden transitions is those that violate conservation of a conserved quantity.  In practice, the only absolutely conserved quantities are the (locally conserved) gauge charges of the electromagnetic, strong, and gravitational interactions:  electric charge, color charge, and energy-momentum.  Transitions that would violate these (super-)selection rules are so obviously forbidden that we do normally even think of them.
The other instance is that there are no transitions between different macroscopic thermodynamic states in the thermodynamic $N\rightarrow\infty$ limit.  This is never the true situation, but it has relevance if you are considering, say, an idealized infinite ferromagnetic domain.  Then there is no time evolution operator that changes all the spins at once; there is no way to change from one ground state to another (even though that is obviously possible for a real, finite system).
The issue in both these cases is that you cannot pick out term in the series expansion of $U=\exp(iHt/\hbar)$* that produced the transition required.  In the first case, the time evolution terms never change the conserved quantity.  In the second case, there is no term in the sum (even though the sum is infinite) that actually contains the infinite number of point operators that would be needed to change the angular momentum at every lattice point.
*The time evolution operator must be correctly time-ordered if $H$ is time dependent, but that is an inessential complication.
