There is another, more circuitous route to the one given by John Heyer:
Let's assume you're finishing your 2nd term and are hankering for a third:
1) Get chosen as Speaker of the House. There are no limits as to who can serve as the speaker; even though the House has never chosen a non-member, legally they could. You could potentially be chosen while still President. Neat.
2) Stop the Electoral College from meeting the absolute majority (270 votes) required to win. The 22nd Amendment stops you from winning this race, so your job is to make sure no one else does. How you do this is up to you, but if you have some popular support, running someone as a third party to split the vote would seem the easiest way. Interfering in some way with the voting process could also get you there, as again, it's an absolute majority.
3) Prevent the House of Representatives from choosing a President from the top three. You're the Speaker of the House, but sadly you (probably*) cannot filibuster this. There are, however, two ways to fail this process:
a) if a quorum of 2/3rds cannot be had or b) if an absolute majority does not agree.
*The 12th Amendment says "the House of Representatives shall choose immediately", which implies you don't have a choice. But who knows, Congress works in mysterious ways.
4) Prevent the Senate from choosing a Vice President (who would usurp your claim because he's the only guy in front of you). Same two exceptions apply: without a quorum or majority, a Vice President cannot be chosen.
5) Congrats, when you leave office, you, as Speaker of the House, are now the next in line for succession. Specifically:
if, by reason of death, resignation, removal from office, inability, or failure to qualify, there is neither a President nor Vice President to discharge the powers and duties of the office of President, then the Speaker of the House of Representatives shall, upon his resignation as Speaker and as Representative in Congress, act as President.
Presidental Succesion Act of 1947
You are now acting as President, despite not being elected. You can wave at the 22nd Amendment as you drive over it in The Beast. Note that it shot itself in the foot when it says:
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.
IF, as naysayers may say, elected was all encompassing, you wouldn't need any of that after "No person shall be elected to the office of the President more than twice," Because the rest is there, we must assume that there are other ways to hold the office without being elected... like succession!
Naysayers may also say: "but the Constitution, through the Ineligibility Clause, prevents being the President and the Speaker of the House:
No Senator or Representative shall, during the Time for which he was elected, be appointed to any civil Office under the Authority of the United States, which shall have been created, or the Emoluments whereof shall have been increased during such time; and no Person holding any Office under the United States, shall be a Member of either House during his Continuance in Office.
Ah, but here's the rub: "Although the Constitution does not require the Speaker to be a Member of the House, all Speakers have been Members." Speaker of the House then, does not grant Membership, and therefore the President is eligible.