Only 3? If we are talking 3 for everything, I would probably have to say 22lr, 12 gauge, and 30-06. If we are talking centerfire rifle rounds...
1- 30-06 This was the first rifle I bought and I got lucky because I didn't know my rear end from a hole in the ground back then. That Savage is ridiculously accurate and turned out to be the best purchase I have made. It will definitely be going to Africa as soon as funds allow.
2- 45-70 with modern loadings. In a Marlin 1895, it is very handy, allows for very fast follow up shots on multiple targets, and puts game down with authority.
3- 22-250 It does a wonderful job turning varmints into red mist and with proper bullets is perfectly capable on game such as white tailed deer. With it's light recoil, it works well for kids.