If you are installing from the original disc, you are installing version 1.0. This version contains the strict SecuROM activation limits. You must patch the game.

When you see a prompt to "re-enter and verify your registration code," it usually means one of three things:

Follow these methods in order. Stop when the error disappears.

[RA3] Starting Issues | Patch Issues - Support Archive - Revora Forums

Do not use this method. Your game uses a different entitlement system. Use Method 3.

Older versions of Red Alert 3 store the verification status in a hidden configuration file. You can manually tell the game it's already verified.