Here is the full text of Section One of the 22nd Amendment:
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.
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As you can see, there is a window of opportunity there, "two terms or ten years." A President can't be elected to the office of the President more than twice, but if she's inherited the office through succession, and held it for less than two years, she can run and be elected twice more.
Thus, Lyndon Johnson was in a position to serve over nine years, if he were reelected in 1968, but he chose not to run again that year. (Harry Truman was specifically excepted by the technical language of the Amendment and could have run repeatedly after his second term, and threatened to do so, in fact, but did not.)
The office of the Vice President is a separate office and a (very technically) separate election, which is why Johnson could run twice more. Some have argued that a two-term President can run as Vice President, and can serve for two years if s/he accedes to the office again. If that person becomes President or Acting President again, the term would automatically run out at the two-year (ten year total) mark, and the next in line would take over, if the former President/VP took over with more than two years left in the term.
Furthermore, there is no term-limit on the Vice Presidency, so Al Gore and Joe Biden and even GHW Bush are all available to run as VP again and run for reelection if they accede to the office (with the two-year reservation for GHWB). Dick Cheney showed to everyone's loss how important a Vice President can be in the presence of an idiot man-child President. For an obvious incompetent like Donald Trump, selecting a former President as his VP would be a wise publicity move, if there were one alive who would do it.
And then there were the legal research efforts of Ronald Reagan, who in his lucid moments thought it would be great if he stayed President. I don't know where the write-up of it is, but Reagan's people tried first to get the Amendment rescinded, then argued that they didn't actually have to do that, but I don't know what the witchcraft was they planned to use to annul the 22nd. It's possible that they planned a two-step maneuver where he stood as VP and then took over after the election.
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