In 1880 Grant considered going for a third term.
I also read that in Chernow. He was not unwilling as four years before. But he insisted that he had to be asked by the party and didn't really campaign before the republican convention. I suspect though that the presure of the presidency would not be good for Grant's health. He had a cancer, but his last year was a constant agony, because of it. It is not unlikely he wouldn't fulfill the whole term. There's an obvious similarity with Roosevelt here.
A big difference in circumstance is that there wasn't a thread of war, a special circumstance for a third term. Because the principle of two term max has been an unwritten law from the beginning. When it was finally written on paper, the special circumstance was used as a legitimazation of Roosevelt's third term. Now if Grant has three terms, there will probably also be an attempt to legitimize this precedent somehow.
No person shall be elected to the office of the President more than twice in consecutive terms, 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 President more than once in consecutive terms. But this Article shall not apply to any person holding the office of President when this Article was proposed by Congress.
Edited Amendment XXII with the assumption that the strange attractors are stronger than the butterflies (same time, same president).