There has been some debate over the years as whether the presidential term limit amendment is lifetime or consecutive terms.
This is not ambiguous:
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 no person constitutionally ineligible to the office of President shall be eligible to that of Vice-President of the United States.
You can't be Vice President if you've already been President for more than a term and a half.
Teddy Roosevelt was elected VP (as a Republican) as McKinley's running mate in 1900. McKinley was assassinated in September 1901, and Roosevelt succeeded him and served out the rest of the term, three years and a bit.
Roosevelt ran on his own merits in 1904, was elected, and remained President until 1909. He declined to run again in 1908, and Taft (R) was elected.
Roosevelt became frustrated with Taft's conservatism, and in 1912 he founded the Progressive Party and ran for President again, against Taft (R) and Woodrow Wilson (D). This schism in the Republicans meant that Wilson won in an electoral landslide. (And this is why third parties don't work in the American system.)
So Teddy Roosevelt only actually ran for President twice, non-consecutively, and lost the second time. He actually served only one full term. If the 22nd Amendment had been around at the time, he would have been forbidden from the 1912 run, because he served too much of the term McKinley was elected for, but he didn't actually run for President three times.