To give up your Social Security number, you would probably have to renounce your citizenship, since merely leaving the United States, becoming a permanent expatriate, etc. wouldn't be enough.

Paying Social Security taxes is mandatory, however. Even if your income exceeds the means testing limits, you are still required to pay Social security (FICA) taxes on the income that falls beneath it.

That being said, there are several groups that are exempt from paying Social Security taxes, based on what I can scrounge up through online searching.

College students who work under the Federal Work Study program. 1 State or local government workers (like teachers, city workers, police, firefighters, etc.) hired before March 31, 1986 and who participate in their employers' alternative retirement systems. Self-employed workers with with yearly net earnings less than $400 Ministers, members of certain religious orders, and Christian Science practitioners may file IRS Form 4361 to exempt themselves from self-employment taxes, including FICA taxes. The picture is more complicated than this, however (see below). All members of certain religious groups, like the Amish, Mennonites, etc. 2 Election workers earning $1,000 or less a year. Minor children with earnings from household work but for whom household work is not their principal occupation.

If you aren't part of one of these groups or you joined them after already being enrolled in the Social Security system (i.e. paid FICA taxes, etc.), you don't have to take Social Security payments, but you are not exempt from FICA payroll taxes. This means that you can't simply become Amish and be exempt if you've already paid into the system.

Many of these answers were listed here, but the only stated source is "Wikipedia," so take it with a grain of salt. I've tried to provide other sources where I could.

Ministers/religious officials

I mentioned that the situation with ministers and/or religious officials is more complicated. Among other things, these officials need to certify that they are religiously opposed to receiving public insurance benefits and they have to file by a certain deadline soon after earning a certain amount as a minister. Furthermore, if you file this exemption and choose/chose to revoke it later on, e.g. by filing IRS Form 2031, during the short time windows when this is possible, you're no longer eligible to file for another exemption. Once you revoke your exemption, you won't get another one.

Also note that this opt-out only applies to income earned as part of your ministry. For non-ministerial income, you're still required to pay FICA taxes even if you follow the procedure(s) above.

If you do opt-out

Clearly, opting out of the Social Security isn't easy, or even feasible, for many people, but if you are outside the system, common sense dictates that you should insure yourself against risks that would usually be covered, at least in part, by Social Security. Dave Ramsey maintains a list of such investments that he recommends ministers and religious officials who opt out of the Social Security system should purchase, but the rationale applies to anyone who isn't part of the system.

Term life insurance that covers 10x your annual income. Normally, your spouse or dependents will receive a Social Security death benefit upon your death, but if you opt-out, your family doesn't have this option. Since the death benefit is a pittance ($255), however, you may want life insurance regardless of your participation in Social Security. Long-term disability insurance. If you become disabled, you won't receive disability benefits, so you'll may want insurance to cover this risk. Long-term care insurance. Similar to disability insurance, you may want additional income to cover you in case you require long-term care, and Medicaid isn't enough. Other retirement savings. If you're not going to be paying money into Social Security, it makes sense to invest that money in some form of tax-advantaged or tax-sheltered retirement account to ensure that you still have some income in retirement. You won't be receiving Social Security checks, but you'll still need income.

Depending on your circumstances, it's may be a good idea to invest in some or all of these instruments anyway (especially tax-advantaged/tax-sheltered retirement accounts), because Social Security doesn't cover 100% of the risks you'll face, but that decision is highly specific to each person/family.