As for Missouri, I think it would be I-24. By connecting to I-57 in southern Illinois near Carbondale (in turn leading to I-64 at Mt. Vernon), it serves as an important route from the South into St. Louis - which, in turn, has connections to many points throughout the state.
(I would also have included I-57 itself for southern Missouri, but because it actually does enter the state, it doesn’t count. I-57 would work for Arkansas, though.)
Also, here’s another example: I-29 for Nebraska, because it connects to Omaha via I-80 at Council Bluffs - not to mention all of the other bridges into Nebraska from Iowa, thanks to its running parallel to the Missouri River along the western edge of the latter state. (In a similar vein, because it runs along the eastern edge of both the Dakotas, I-29 also serves as a connector for various points in western Minnesota.)