The 375HH is by a large margin the most popular for all Big Five where Buffalo is concerned. After that 416 rem Mag. Then the field gets muddy with so many really wealthy people who buy a gun just for that one hunt. Double rifles, and custom bolt rifles,....... it's not easy to say. The Americans like the 458 Lott on CZ and several other custom mauser actions has become a real popular rifle as well. The Europeans bring a 9.3 frequently as well.
For Plains game in my camp and everyone I have ever worked in the 300mag in one cartridge or another is used by more hunters then all other cartridges combined. I don't think I have ever seen a group of American Hunters without at least one or two of them. Of the 300 mags Americans bring the 300win and Weatherby are the two most common. Then the 300HH. Of the non- American hunters the 300HH is more common.
The Dangerous game hunting has become quite expensive. To the point a guy could hunt for 2 weeks and shoot a pile of plains game for much less money. Buffalo regardless of what you see posted as a "special" will be in the 10K range or more after you pay every nickle and dime. Elephant and lion will be pushing 20-25K, Leopard also in the teens for the most part. Rhino is gonna be in the 20's as well. Some of these will be less when done in one hunt but the daily fees alone are going to be over 1000 bucks a day for a 21 day multiple species dangerous game hunt. That puts you at 21K before air charters and trophy fees. Heck on these hunts the tips are nearly as much as the plains game hunt would be! I guess that is why so many pay the extra few bucks for a custom rifle, it's just pennies compared to the whole trips price!