I came across this question in an introductory physics course awhile back and I never got over it: "A hydrogen atom has an electron in the n=5 orbit, what is the maximum number of photons that might be emitted so that it decays to the ground state (n=1)?" The answer the professor was looking for was 4 since he was picturing the atom going $5 \rightarrow 4 \rightarrow 3 \rightarrow 2\rightarrow 1$. This sort of answer is how we are taught to think of atomic transitions in introductory physics courses, but I can't imagine its the complete picture. As far as I can tell the only constraints on the system are energy and angular momentum conservation and if you produce pairs of photons with opposite angular momentum you can produce infinitely many photon pairs, and thus infinitely many photons so there is no max number. There are obviously 'selection rules' on these sorts of transitions but they always seem to procede by some dominant process when as far as I can tell there can be higher-order processes that allow many more transitions to occur.
In short: given the full machinery of quantum mechanics, is the answer to the professor's question even finite? Is he right? Or is the number potentially infinite?
Note: I acknowledge this is a exam/HW problem, but from a loooonnnnngg time ago, and I'm looking for the 'real' answer, not the fake introductory answer, if there is a difference.