A good thread; I have been pondering about the same question many a time.

I certainly subscribe to what many others stated: often people tend to use too much gun on a safari.
On the other hand, the standard moose caliber in Finland is too small.
(I know - these are not DG, I will come to that.)
I have the very extensive experience of 2 (two) PG safaris. On both trips I used a .30-06 and on the other an 8,2x53R, too. Both calibers performed excellently, the .30-06 taking an eland bull @ a lazered 176 meters (199 yds) with a single shot.
Now, especially this experience plus the wide use of the 6,5x55 as moose rifle has made me think how small can you go.

If the .30-06 took the eland so nicely, would not a 7mm have done the same job? Probably so.
OK, so how about a 6,5mm? Probably so.
How small a gun could have done the same? I don't know.
Was the .30-06 enough gun? Apparently, it was as it did the kill. However, I still do not believe it was a sensible caliber to be used. Best would probably have been e.g. a .338 (mag) or 9,3x62 or thereabouts. I have no scientific way of proving this at all.
My hunting buddy and the PH strongly maintained I should have uset the .375 H&H and the director of the outfit nearly didn't believe his ears when he heard I had taken a one-shot kill on eland with a .30-06.

In Finland, the most common moose caliber is the .308 (180grs bullets). Probably more yet have been taken with the 7,62x53R i.e. essentially the same power.
However, given the size of the moose I firmly believe the most suited is the 9,3x62 or comparable. The moose is big and strong although not dangerous.

How about DG?
Over the past ten years or so, about once in a year a hunter has been attacked by a bear in Finland. The hunt may or may not have been for bear. In most cases, the hunter has fired in self-defence. Now, being mauled by a bear means the shot did not stop it. Why? Too small a caliber? Bullet failure? Poor or missed shot? All of the above?
I don't know but what I do know that these have been cases where the hunter used a .308 (maybe not in all cases, but in most cases).
I discussed the matter with a very experienced bear hunter who listened to my theories and nodded his head. On my question on suitable bear caliber he responded: "If you have a .308, by all means, don't buy anything smaller." He uses a 9,3x62 and his hunting partner a .45-70.
The 9,3 has gained more ground again as bear hunting has become more common.

For sure "overgunned" one cannot be in the sense that an animal cannot be killed too dead. However, it is possible to be undergunned.
Probably the only "absolute" limit I would set is that for elephant, the minimum is the 9,3x62 (or if rimmed, 9,3x74R). I.e. a .35 Whelen is too little, a 9,3 is not. The line has to go somewhere and I think it goes here. "I think" means I don't know nor can I substantiate this view with any hard data.

I think that I would maintain that the sensible minimum for DG is, broadly speaking, a .338 magnum.

Of course it all boils down to the hit and bullet performance.
A large caliber cannot compensate for a poor shot; and a small hole in the right place will bring instant death.
Considering the fact that a dangerous game animal may attack you, you will need good initial knock-down power, and especially if you're attacked you'll need some stopping power. This is brought by larger bore and heavier bullet - probably more so that high velocity.
Thus, a sensible limit can probably be discussed in an educated manner.

I certainly would not go bear hunting with anything smaller than a .30-06, and I would opt for something larger - 9,3x62. The same goes for lion, elephant, buffalo and so on.

If you look for a guidline as to "when does a caliber become too small" I don't have an answer.
Sensible minimums are set by game laws, PHs' recommendations and more experienced hunters' views. Put these together and for each animal you'll find the line.
As a closing comment, though, I would not too use too much gun, either. This results in poor shooting and too much recoil for a follow-up shot. Having never hunted elephant I'd say my maximum limit goes with the .416 Remington magnum.

- Lars/Finland