That would be the Twenty-Second Amendment:
Originally Posted by CanadaJohn
Section. 1. No person shall be elected to the office of the President more than twice, and no person who has held the office of President, or acted as President, for more than two years of a term to which some other person was elected President shall be elected to the office of the President more than once. But this Article shall not apply to any person holding the office of President when this Article was proposed by the Congress, and shall not prevent any person who may be holding the office of President, or acting as President, during the term within which this Article becomes operative from holding the office of President or acting as President during the remainder of such term.
Section. 2. This article shall be inoperative unless it shall have been ratified as an amendment to the Constitution by the legislatures of three-fourths of the several States within seven years from the date of its submission to the States by the Congress.
It seems to apply only to one being elected directly to the position of President. I imagine there could be some fights in the courts over it if it arose, but by my reading, it doesn't seem to prevent one from being elected or appointed to some other position that puts him in line to the Presidency, such as Vice President or Speaker of the House.
I do think that Clinton/Obama ticket would be very polarizing. I think there's a very strong correlation between those of us who are fed up with the incompetence, bad judgement, hubris, and outright corruption that Obama has represented; and those who think that Hillary Clinton would just be more of the same. It's generally a good practice among nominated Presidential candidates to pick a running mate that has strengths to cover for the weaknesses of the main candidate. Mr. Obama would only aggravate and underscore Mrs. Clinton's weaknesses.
Before any Americans answer, John, do you think you can name the one President of the United States who achieved this position without being elected to it, or to any other position close in line to the Presidency? As far as I know, it happened only once.