(Updated 11/16/2021.) It is common for two different US routes to overlap (meaning they are concurrent along the same highway for a certain distance before splitting apart again). It is less common for three (or even more) US routes to overlap. For example, traffic heading west from Cody, Wyoming, towards Yellowstone is on US 14, US 16, and US 20 (all at the same time).
Sometimes the two routes are signed in opposite directions. For example, heading west from Wytheville, Virginia, a short segment of interstate carries not only northbound I-81 and southbound I-77, but also northbound US 11 and southbound US 52. This arrangement is sometimes referred to as a "wrong-way overlap", or a "which-way overlap". These are a little more rare, but nothing to get too excited about.
A few more instances can be found in Oklahoma. This is in Seiling:
Another Oklahoma example is to the north of Enid...
...and another is between Ponca City and Tonkawa (if necessary, click the + button to zoom in, and then pan):
Are you aware of another example of a three-way wrong-way US route overlap? If so, please comment and/or send in a photo.
Or how about a single road that carries all four cardinal directions? I'm not sure there is such a thing, but this photo shows a sign assembly that comes close. Not only are there three different highways with three different directions, but two of those highways are directionally-suffixed, so all four cardinal directions are referenced on this signpost: