President Roosevelt served an extra half term when he took over while serving as Vice President when Truman was assassinated and he had to take over as president. Would it be possible for a Vice President, to in theory, serve infinite terms as president by continuously taking over when his president died/was assassinated?
No one can be appointed to the office of Vice-President who is ineligible to hold the office of President.
The 22nd Amendment states "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."
So /u/No-bats is correct, and once you've served your terms you cannot be chosen again as Vice-President.
I believe how it is structured, he could serve 10 years. Two full terms and half of one if he was VP and the President was killed/died. But if he served 3 years of a term, he can only run for one more term.
BTW, Truman was not assassinated. Roosevelt died in office. He ran for 4 terms. Before that, no one ran more than 2. It was an unwritten policy/tradition as George Washington only served 2 terms. After Roosevelt, they enacted an amendment to limit the amount of terms you could serve.
Not quite the right place for this question, as it's a civics one.
However, due to the "history" based nature, I'll allow it.