As many as will fit in the amount of space available on the card.
J Charles Ferrari Eng & Nishimura 213.622.2255 The statement above is general in nature and does not constitute legal advice, as not all the facts are known. You should retain an attorney to review all the facts specific to your case in order to receive advise specific to your case. The statement above does not create an attorney/client relationship. Answers on Avvo can only be general ones, as specific answers would require knowledge of all the facts. As such, they may or may not apply to the question.
Generally speaking, the name on your green card needs to match the name on your birth certificate. The exceptions are if you are married and took your spouse's last name or if you had a court-ordered name-change. There is no limit on the number of names, but the name should match your birth certificate.
This general advice does not create an attorney-client relationship.