In 1908 Republican presidential nominee William Howard Taft offered the vice-presidential nomination to New York Governor Charles Evans Hughes. Of course, technically the nomination was the convention's to decide, but had Hughes accepted, the convention would probably have gone along; as in 1900, the New York GOP bosses would be glad to get rid of a reform-minded governor. But Hughes declined:
"Taft offered the vice presidency to Hughes but he turned the place down. 'It would be impossible for me in any event to take the nomination for Vice President,' Hughes replied to the White House." Lewis L. Gould, The First Modern Clash Over Federal Power: Wilson Versus Hughes in the Presidential Election of 1916, p. 46.
Suppose Hughes had accepted? The Taft-Hughes ticket wins. I doubt very much that Taft will offer Vice President Hughes the Supreme Court seat he offered Governor Hughes in OTL in 1910. And if he doesn't, Hughes as a matter of loyalty will almost be forced to back Taft in 1912 (which wouldn't be that hard for him, anyway--he hadn't gotten along particularly well with TR). But that destroys the major rationale for nominating him in 1916--that as a Supreme Court justice he had been neutral in the 1912 Taft-TR fight and could therefore unite Taftites and Bull Moosers behind the 1916 GOP ticket...