Is this possible using some clever bitcoin script? The intention is to get more privacy. I can give different addresses to different people still no one can guess how much I make in total. I can do this by generating several key pairs but I will have to save all the private keys somewhere. If I can come up with a way to generate multiple addresses, I will have to save guard only one private key.
One private key maps onto two addresses - one that uses an uncompressed public key, and one that uses a compressed public key.
What you might also be interested in is a deterministic wallet - a Bitcoin wallet holding multiple addresses generated from one private key / secret. This will let you secure multiple addresses and store only one key.
Besides the HD wallet technology, you might want to investigate "stealth addresses" that are new to Bitcoin. Not aware of any easy to use wallets supporting stealth addresses yet, unless you are into the Darkwallet.
The libbitcoin framework currently supports stealth address capability for Bitcoin, in particular see the bx command which incidentally is deprecating the sx command also mentioned Andreas Antonopoulos's book - available at O'Reilly. bx is pure C++ code while sx was a combination of C++ and Python.
I suspect the airBitz wallet, currently HD-enabled will be one of the first mainstream mobile wallet applications to support stealth addresses because it is built up the libbitcoin foundation.
I believe the one major issue with stealth addresses is both the sender's wallet and the recipient's wallet must support it because the address formant is very different from the typical Bitcoin address. (Have not yet experimented with BX enough to say if this is true for certain.) Stealth addresses are effectively performing Diffie-Hellman-Merkle key exchanges behind the scene. More information must be conveyed to make this happen.
In a nutshell, Peter Todd's stealth address invention will be great to print on business cards to share while still maintaining your financial privacy. Only waiting for the stealth address BIP63 to be posted at https://github.com/bitcoin/bips. Not sure what the politics are...