I spend more time thinking about this than I should, and for no good reason
The way I look at it, the question was for rifles, meaning a firearm with rifling, so a smooth bore shotgun is not included. Unfortunately he didn't specify centerfire or rimfire though

If it was what 3 centerfire rifles, that'd be easier: 223, 30-06, 375.
I chose 223 over something more "varmint" like a 22-250 or 204 because it can also be used for target shooting in disciplines such as f/tr or service conditions, while still being a very capable varmint round.
The 30-06 is the middle ground for cxp2 and cxp3 game. With a 22" barrel and 7.5-8.5 pound rig all set up with a scope somewhere around 2.5-8, 2-7 or 3-9, it will do well as a mountain rifle, brush rifle, it will do passably on just about any cpx2 or cxp3 game in hunting ranges under almost any conditions. Perfect for all, no, but serviceable. It can also pass as a backup emergency varmint rig with 110gr bullets or an emergency backup for the big bears with 200+ grain bullets. I chose it over the 270, 280 or 7x57 because I value performance on bigger animals and maybe something that might trample or bite you more than I value slightly flatter shooting or slightly less recoil. If one wasn't worried about having a backup a 270 would probably be even more perfect for this role, but as I said I like contingencies.
The 375 will be a good choice for anything from elk size on up, including African dangerous game. Perfect for elephant? Once again not perfect for everything, but serviceable for anything. Can serve as a backup deer gun, believe it or not.
But if the OP is including rimfire rifles in the limit of 3 that throws my who idea out

The way I see it, you just can't be without a .22lr. 80% or more of my practice is with a 22, which I think is how it should be. When it comes right down to it, losing out on the inexpensive practice and the small game hunting ability is too much to lose so I would have to give up one of the above three... I guess it would have to be the 223... but I don't like it.
22lr
30-06
375