Looking at this question, I've been wondering, What are the biggest private companies which work in both animal and non-animal selective breeding?
Let me know if Biology.SE is not appropriate for this kind of question.
This varies by the kind of animal or plant. I think one company would rarely do both.
For plants Monsanto comes to mind. And Syngenta.
I'm not sure for animals. For beef cattle in the US, this is done by individual ranchers (who may be corporations) that collectively share their genetic data creating a market which can predict the value of a bull. The market for bull semen can be worth hundreds of dollars per impregnation.
Why are bulls valued at such a greater rate than cows? Because semen can be preserved, transported and used thousands of times while a cow can only calf one at a time.