The .338-06 fairly duplicates the old .333 Jefferies and that was considered a fine "medium" round for elephant. Given that, plus modern bullets and a touch more velocity, I doubt theres any large game in the US that the .338-06 wouldn't cope with..

That said, I went with the 9.3x62mm as it seemed to be more available this side of the Atlantic, and after taking a dozen critters with it, I couldn't be happier...