The varmint to big game cartridge will be a compromise that will not fill any particular niche as well as a more specialized piece, but then that's the attraction isn't it. I approve of the concept of the single rifle that can be carried day in and day out, able to hit a mouse with one shot and kill a moose with the next. The more the rifle is shot the more intuitively you can shoot with it, and the more confident you become. The difference between the competent marksman and the outstanding marksman is often nothing more than confidence. The more you confine your shooting to a single piece, the better you are apt to become with it.
Rather than cartridge choice, the real secret to versatility is handloading. Almost any modern cartridge can be loaded to effectively solve a wide variety of shooting problems, and most rifle cartridges from .25 to .40 have their champions. I for example reach for my custom .375 Ultra carbine more often than anything else, it is as accurate as a varmint rifle, shoots as flat as a .300 magnum, cast bullets are fun to shoot and put small game in the pot, and loaded with heavy bullets at full steam it is enough for any land animal on the planet, and many marine mammals as well. That was my choice, but you must determine the parameters you need to cover and make your choice accordingly. But keep this in mind, it is the package you search for rather than the cartridge.
The rifle should be light for portability, and short for quickness. The stock should be immune to changes in temperature or humidity. The sights should be equally useful at long range or short, and it would be wise not to ignore the addition of irons to a mid-power variable scope. The rifle should be accurate enough for the most demanding shooting you will call upon it to do, but this is a lesser problem as it seems that today's rifles which do not shoot MOA with a wide variety of loads are the exception to the rule if the range reports are to be believed.