I was driving through the White Mountains today on ME-113. The road actually runs through New Hampshire for a while, but is still signed the whole way as Maine Rt. 113. Does this happen anywhere else?
113 is actually Y shaped. Coming south from Gilead, ME to Fryeburg, ME when you hit US-302 it splits. ME-113 takes a left continuing on towards Portland and NH-113 takes a right to go over the top of Lake Winnipesaukee to Ashland, NH
Interstate 95 passes through a small wedge of Washington, DC (outside of any specific US state, by definition) that juts across the Woodrow Wilson Bridge. This phenomenon is not marked in any way by signs along the road. Eastbound, there is a Welcome to Maryland sign. Westbound, there is a Welcome to Virginia sign.
That means that if you have traveled up the length of I-95 from Virginia to Maryland (or down the other way), even without stopping, you can legitimately claim that you have traveled to DC.
Is there actually a ME-113 sign in New Hampshire or does the highway just very briefly become NH-113?
It’s not all that uncommon for state highways that link up with each other to be similarly numbered in spite of the states’ normal numbering schemes. One very long example is there’s a chain of State Highway 200’s that runs from Sandpoint, Idaho through Montana and North Dakota to just shy of Duluth, Minnesota.
It stays as ME-113. There is another road nearby called NH-113, but they don’t link up.
That’s an Interstate, though, not signed as any particular state, so not really what this thread was asking about. Plenty of Interstates go out of one state into another without changing signage. That’s what they do (sometimes anyway, there are some that don’t like Hawaiian Interstates).
This happens with Maryland Route 404, which becomes Delaware Route 404 when you cross eastbound into Delaware. The corn and tobacco fields don’t seem to care much.
There is even at least one international example of this, where US Interstate 95 (I-95) becomes New Brunswick Provincial Route 95 after you cross the border.
But Interstate route signs used to and in many places still do have the state printed on them.Like this:
Hm, I thought that the northern route into West Yellowstone might count, but it turns out it’s numbered as a US route, not MT.
If google is to be trusted, California State Highway 161 is really in Oregon (by a few dozen feet or so) for a few miles: Google Maps
There’s a bunch of those in BC, like US-97 which turns into BC-97 and eventually becomes the Alaska Highway (although the 97 numbering goes away when it goes into the Yukon.) Also BC-99 even though US-99 doesn’t exist anymore. And weirdly BC-101 which runs along the Sunshine Coast is obviously supposed to reference US-101 (which mostly runs along the Pacific Coast) but the two highways aren’t anywhere close to linking up.
New York route 17 (which is becoming I-86) dips into South Waverly, Pennsylvania for about a mile. It keeps the New York signs and is maintained by the New York State Department of Transportation.
Considering how well PA maintains its roads, that stretch of route 17 is much better off being maintained by NY. 
Add Ontario’s 61 to the list. It starts in New Orleans and ends up the road from me in Thunder Bay. The next border crossing to the west at Ft. Frances has a continuation of Hwy 71, which originally started in Baton Rouge, and intersected 61 in Louisiana before heading north on its own. It’s also a bit of a mind-fuck, for Ontario 17 (our longest) and Ontario 11 (our second longest) share the same pavement for a couple of hours including where 61 intersects with it, so I’m used to 11 being 11/17, only out Fort Frances and Kenora way it is 11/71. Rocks, trees, water, 11/17, rocks, trees, water, 11/17, rocks, trees, water, 11/17, rocks, trees, water, 11/71 – huh?! – makes ya wonder if you need to pull over for hallucinating. Anyway, 11 continues from Rainy River, Ontario, through Minnesota all the way to the North Dakota border at the Red River south of Winnipeg.
I once drove up Jutland (Denmark) to Friedichshafen on highway E4, took a car ferry to Goteborg, Sweden that had a giant E4 sign painted on its side and then drove off on E4 in Sweden.
Heading south on Missouri Highway 43 you eventually reach the MO-AR-OK corner of the state. At that point for the next >5 miles the road is signed as*** both*** Arkansas 43 and Oklahoma 20.
I’m picturing Robin Williams speaking with a southern accent.