Gmail – Change Gmail Contact defaults

There are several options available:

• This form of verification only happens occasionally, so you could try creating another account.
• You could try filling out the support form linked to on that page and select "I don't want to give my telephone number to Google" as the issue. If enough people do this, it may persuade Google to use some other means of verification. I'm not sure if they will activate it for you or not since I went with the last option:
• Choose to verify by voice rather than text message, and enter in your phone number. While this does give them your phone number during the sign-up process, it seems like it isn't stored in your account. You can go to Google Account Settings > Change Password Recovery Options, and you will see that there is only an SMS section, and your phone number isn't stored there. It might still be stored in Google's database behind the scenes though.
• If you are transitioning as part of the Google Apps upgrade process like I am, and your old account had Google Voice, you could enter its phone number, and Google will send the verification to that number.

Update: even if you verify by text message, the phone number won't appear in your settings.

One easy way would be to use a disposable email address via DisposeMail.com as the alternative email address Google registartion asks. That way you don't have to register to any other site, and you could even use the disposable address to recover your account just by writing down the address somewhere - no expiration.

Of course, it's better if the disposable address is nothing somebody else might use, for example an SHA1 hash of something. Like 0db939e32d0cc9a49f2660968676c8692b114713@disposeamal.com

And if you want to be really, really anonymous you could use a bootable USB-stick for a separate, secure OS to protect from tracking methods like those used by Panopticlick, and of course route all traffic thorugh Tor... but I'm thinking you don't need quite that depth of anonymity. :)