"Of all time..."; is a big order to fill by any cartridge...
If sheer numbers of "kills" were the criteria... the list would "hands down" include the 12ga and the .22LR. Without a doubt, the vast number of "varmint/predator" kills were, and still are, opportunistic events. For all of my youth there was a .22LR and a 12ga. behind the kitchen door of virtually every farm house in rural central Illinois. They were there to protect the chicken house and barnyard, and they got used! Farther west the .44-40 and the .30-30 replaced the .22 or was added to the list. But, "kills" alone are not the criteria or we would probably have to include the "Ford" (or some other automobile/truck brand).
I think that it has to be defined more in terms of "effective longevity". A kind of "which one has been doing it well for a long, long time", to be recognized as a "greatest cartridge... of all time". And, for our purposes it is probably not focused on opportunitic kills, but rather on purposeful "hunting" of varmints and predators.
Historically and up to the present day, a long long list of cartridges have had their moment in the sun... The .218Bee the Wasp... even the .244 Remington (and my favorite cartridge of all time the .257R) have all been "stars" on the stage at some point in tme. But, it is like this... in my opinion, who started out "way back then", and from among that list, who is still on the stage, maybe not the star of the moment (the .204 certainly has its great start recognized, but will it be there in another 50 years?).
So, here goes, first on my list is the .22LR. It is a "been there and done that" cartridge, and is at the top of the "greatest of all time list" of mine. First of all it has been around and doing a job from almost the day we switched from rocks to guns. Second, I bet the vast majority of us shot our first "varmint/predator" with a .22LR, and own a darn fine one right now and include it on our list of "have to own" rifles (and pistols also).
Second, the .22 Hornet. Again, it was "(re)invented" about 1930 (a modernized .22WCF) and is still commercially available and efficiently doing the job within its limitations. 75 years of being on the stage as one of the chosen is proof enough of its deservig of recognition. Come on, admit it; you either have one or want one and you know it "works". Besides, it is a sleek looking little cartridge and factory rifles come in such "cool" looking actions! This one also gets a lot of "style points" for nostalgia because to a large extent it is the "mother load" (nice pun, huh?) of varmint hunting. It was introduced to cater to a desire for a specific cartridge and rifle combination to appeal to a hunter who was specificlly targeting "varmints".
Third, a round from about 1950 and still enjoying some popularity, the .222 Remington. (I am aware that this is a tough call and might be the most controversial inclusion on my list). No question, that some newer cartridges can do it better and this one my be later eclipsed by another cartridge of more moden vintage and get pushed off my list. But, for now I must include it. It was and is such an outstanding performer in the accuracy departmnet that it would be included on that basis alone. My own, a 1968 vintage Sako Vixen will still produce an occassional .35" group with these old eyes and hands. It has, for me and countless others killed more "varmints and predators" than any rifle I now own. I do not feel handicapped until I have pushed it past 300 yards, and that doesn't happen for me very often. I also know that better shooters than me can get it to do well past 450-500 yards. So, for now it sits in third place.
Fourth, another .22... The .22-250. This is probably by sheer numbers, today, the most popular varmint cartridge or at least tied with the 5th. It fits every criteria of greatness I can conjure up. It is accurate out to incredible ranges. It is effective out to the same incredible ranges. It is easy to load and relatively forgiving in choices of powder and bullets (fools like me are successful with a lot of "concoctions", so it is fun to own). As many more years pass, this round is likely to become the "greatest" of the "greatest", if not already so...
Fifth, is a "tie". (Cheap way for me to get to have six cartridges...LOL) But, these two are in my opinion tied at the moment. Maybe one or the other (or both) will replace one of the first four. So, "Fifth Place I" goes to the .223. Ballistically it is not quite the .22-250 but it is superior to the .222 and still extremely versatile. And, chambered in all of the "AR" rifles it has to be included. It, too, fills all of the requirements; accurate, efficient and effective. "Fifth Place II" goes to the .243. While not as efficient (in my book that does mean "cheap" to load), it is accurate and effective. And, if the "beware of a man with only one gun" rule has applicability, that guy might want to be armed with a .243. You can scratch off "varmints" at incredible ranges in the morning and sit a deer stand in the evening without changing rifles. That is a big plus. It has been around snce about 1955 and still sells incredibly well. Can you say, "longevity" along with "versatility"? That there is an incredible array of bullets available to loaders shows that the bullet manufacturers share my respect for this cartridge.
And, oh how I wish it had been "ten"... Honorable mention (in no particular order) goes to the .22 Magnum Rimfire, the .220 Swift, the .25-06, the .257 Roberts and, the .221 Fireball (for bringing the pistol into the 20th Centiry as a long range Varmint Hunter's tool).
Alas, one final comment, opinions, like "you know what" are a commodity owned by everyone... And, like its counterpart, it might just be producing "####"... So, dn't think that this long dissertation is anything more than the distilled experiences of a very fallible thinker. But, what a great question.