Well, There has been a lot of favourites, but to begin at the beginning...
My favourite cartridge was the .308 for about 20 years because it was just a tad better at everything than the .303 British that I shot my first big game animals with. For many years and many animals the .308 did all I could ask, and then some. It still works, and would be my choice if I was limited to only one cartridge to hunt Canadian big game, but where's the fun in that???
Cartridge favourites are heavily influenced by rife type. As I explored classic European rifles and cartridges, the 7x57 became very prominent. A Brno 22F 7x57 soon rose to the position of favourite, not because it was better than the .308, but as the .308's equal, the combination of the 7x57 cartridge and Brno rifle was just so elegant, stylish, historical, and capable. Then I became enamoured with combination guns and drillings. There's not point in chambering a break action rifle in a rimless cartridge when a rimmed cartridge is better in every way. So I became a fan of the 7x57r.
But European hunters have long since passed over the 7x57 in favour of the 7x64. Wow. Love it. Versatile, fast, powerful, low recoil, and has the panache of a somewhat exotic choice. My favourite in a few different rifles. It's everything the .280 Remington should have been the first time, but Remington screwed up that one and in comparison the .280 has no real reason to exist. The 7x64 did it all first, a couple decades earlier, and better.
But then bigger game called. Nothing more traditional than the .35 Whelen for elk and moose and bears and such critters in North America. I've taken a couple dozen elk and a few moose and several other big animals with a couple different .35 Whelen rifles. Very, Very decisive even when pounding bullets through big humerus bones and into the vitals. Never a failure if i did my part. I like that a lot. Powerful, predictable game killing performance, with manageable recoil. I don't desire any more "long range " capabilities, it delivers what I need. The .35 Whelen became a loyal favourite.
But then Africa called. No proper African hunter shoots a .35 Whelen when he could be using the first, best , and most capable medium bore of all, and the one with the most worldwide acclaim, the 9.3x62. After taking Kudu, and Gemsbok, and Giraffe with my 9.3x62, and some Elk back home, I came to appreciate the cartridge as a "new to me" classic choice. It is a current favourite.
And oddly, the one cartridge that I have hunted with the least but obsessed over the most has the qualities of a favourite that I have passed over when it came to the final choice. The .375 H&H is the one cartridge that is universally acclaimed as the most capable and versatile in almost any situation, for almost any game, around the globe. I have owned and shot 5 different .375 rifles, each one was a marvel and a very capable tool. I still have two. I have used them to successfully hunt black bear and moose and elk and whitetail, but sadly never Cape Buffalo or any other dangerous game.
I chose a Merkel double rifle chambered in .450-400 3" Nitro express for the buffalo. Pretty specialized, but i like it a lot...