I have an interesting barreled action - Ruger No.3 with a pretty heavy barrel in .375 cal.
The chamber is huge!
A rimmed cartridge 3-1/4" in length
.570" base
.562" shoulder 2.75" to shoulder (short shoulder like .160")
.250" neck

Really, it looks like a 375 based on the 470 nitro express case with a steep shoulder.

I haven't been able to pin it down
Anybody see anything like this before?

could it be an improved and necked down 500/416 Nitro express? That would be my best guess.

500/416 is a three inch case though.
The issue is that I have little interest in shooting this and the far forward shoulder preclude it from being rebored and cut to anything else without setting the barrel back quite a bit.
Was thinking a rebore/chamber to 450 No.2 Jeffery would have been possible but the shoulder won't clean up without setting the barrel back.

Sorry, was thinking of the 3.25" version of the 500NE. Do you have dies for the round in question? If not I would just do a re-barrel. Anything that will clean up that chamber is not going to be something fun to shoot out of a no. 3 anyway.

No dies.
Really don't have that much ambition for a project like that.
The barrel is quite heavy and long so even a long 500 or a 500 3" with a long throat would be manageable I'd think. Even then, the shoulder dia. may be a problem.
Probably sell it to a more ambitious rifleman...

Here's a chamber cast - sucks a bit...

What's the rim diameter?

Looks like .660" rim cut - .657" case rims fit in nicely.

I did a .416 on a 40/90 Sharps Straight, the .375 version would be a fine one also.

That is a lot of case! I can't venture a guess but I have a No.3 Ruger and I don't want to shoot that cartridge in it.

Looks like a big mistake someone made; this "cartridge" would be a totally bad idea, way overbored, and in a Ruger #3, which weights like 6 pounds, completely unshootable.
Throw the barrel away and put something useful on it.

Sounds like bubba got carried away. I'd be tempted to set the barrel back and chamber it for 375x2.5. Slim up the barrel and it might make a nice woods rifle. I hate throwing things away.