Private companies don't have to be corporations at all. If investors don't like the "double-taxation" problem, then drop the incorporation and use a different entity. Of course, they would loose the benefits of incorporation (i.e. limited liability) -- clearly many investors find that benefit worth the tax cost.
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If a corporation doesn't want to pay corporate taxes, it always has the option of going private.
On the off chance that you don’t understand the downvotes: all corporations, public and private, pay taxes on recognized income.
Private companies don't have to be corporations at all. If investors don't like the "double-taxation" problem, then drop the incorporation and use a different entity. Of course, they would loose the benefits of incorporation (i.e. limited liability) -- clearly many investors find that benefit worth the tax cost.