What it says in the title, with a POD starting with Andrew Johnson's first day in office, give Grant 4 terms in a row.
He has three big things going against him in 1876: corruption, the Panic of 1873 (technically, the economy had stopped declining by 1876, but that is like saying the Great Recession was already over in 2010--unemployment was still far above what it had been); and the anti-third-term tradition. Maybe he could have avoided some of the scandals, but I don't think anything in the economic orthodoxy of that day could have prevented the depression, and of course the anti-third-term tradition was well established by then. So I don't think it's very likely. After 1874 the Democrats controlled the House, and if he tried to retain power by using troops to prop up Republican state governments in the South, they had the threat of withholding funds. Besides, even if he could get a couple of more southern states in the Republican column by using troops, that would just cause a backlash in the North.
It was only by nominating a candidate (Hayes) who was thought of as something of a reformer and free of the taint of "Grantism" that the Republicans were able to win in 1876 (it also helped that he was from the key state of Ohio, but he nevertheless just barely won it) and even then they lost the popular vote, and only won the electoral vote through very questionable means (not that the Democrats were by any means innocent, of course).
No doubt you have FDR's four terms in mind, but there wasn't any Hitler in the 1870's...