Can you legally denounce your social security number?

You can take an ad out in the paper saying "I denounce 123-45-6789!" in bold type and it's perfectly legal.

That's the ONLY way in which the question makes any sense to me.

If you mean that you don't want to participate in the social security system at all, there is one option I know of: move to an Amish community and become Amish. The Amish are permitted to exempt themselves from Social Security taxes on religious grounds, because the Amish community has a very long history of taking care of elderly/infirm and destitute members (who therefore don't collect Social Security benefits either). You've got to really be committed to the belief system and the lifestyle to do this, though, because if you try it as a tax Dodge both the US and the Amish will take a very dim view of it.

The other option is to choose not to participate in social security by never applying for benefits. I don't think anyone can make you take social security benefits if you don't want them. You'll still have to pay the FICA taxes though.