It would grow more slowly, but could not be eradicated completely, unless you managed to take away military grade cryptography from the people. That would require turning the US into a draconian police state, and even that might not be enough, as the example of Iran shows.
If individuals manage to maintain access to the internet and the use of military grade open source cryptography both on the internet and offline, then this will have tremendous consequences, both good and bad. Cryptography, like nuclear weapons, will change the rules of the game forever, whether we like it or not. It will enable totally untraceable digital cash. David Chaum has shown this is mathematically possible, and the fact that Bitcoin has shown a grassroots P2P open source currency is possible, makes it inevitable. Also, just like nuclear weapons, cryptography cannot be uninvented, and by the Streisand effect it cannot be taken off the internet without shutting down the entire internet. At a minimum, governments will be just as unable to eradicate Bitcoin as they have been unable to eradicate the drugs trade and money laundering.
So, cryptography has great consequences, and it will take draconian repression to try to stop it, and in the end, people aren't going to accept draconian repression. Nevertheless it's going to be a long, hard struggle to maintain our liberties. Society doesn't yet realise that it will have to choose between tyranny and radical freedom. The power of cryptography does not allow for a compromise position. It's not so much that more reasonable compromises cannot be thought of, it's that they cannot be enforced. Any enforcement action that can counteract the power of cryptography requires draconian repression, and is therefore by definition incompatible with a compromise position. See
http://osaka.law.miami.edu/~froomkin/articles/tcmay.htm.
Nevertheless, politicians and society at large will instinctively seek a compromise position, and it will take time for them to realise that there isn't one. Unfortunately, that means that they will initially be unwittingly supporting the move towards tyranny, until they finally realise what's going on. What we in the Bitcoin community can do is to try to speed this process along. The sooner people realise that the world has changed, the less collateral damage there will be.
