The big issue is scope elevation. There is nothing that you can legally shoulder fire that isn't going to need some serious scope base adjustment to reach those distances.
I have a mechanical scope base that gives me around 225 mins of up. I am working on getting a bit more. Then you have whatever the scope will give you.
I have shot my 6.5 mystic pushing 139gr Lapuas to 2300yds accurately. With the extra elevation, I am hoping to get out to 2600yds or thereabouts.
Yes, it gets bounced around alot but if you can get steady conditions, accuracy is decent. Will try some 140gr Bergers this spring.
The biggest hurdle you are going to run into is bullets. Some bullets will go subsonic and stay accurate/stable, others will not.
There isn't a single sub 50cal bullet that will not go subsonic at 2000+ yds. So all the paper numbers in the world will not help if your bullet tumbles. Unfortunately, there are precious few shooters who are ACTUALLY shooting that far to give feedback on stability and accuracy.
At the Summerland LR shooting this spring, we will be pushing a variety of large and small cals out to 2500yds, maybe a bit further (I think we run out of dirt at 2700yds). We will know in graphic terms what will make the trip and what will not.
The 223 and 308 will make the trip, given the right bullet, but the amount of up required is so high, it becomes more a mechanical issue then ballistic.
To answer your question directly, 6.5 Mystic or 260AI is the smallest I have used. The 7mm 180gr Bergers would be interesting to see if it works. Pretty sure the 162gr Amax can be made to work.
Then the various 30's and 338's.
We know that many of the heavies in the 30 (220, 240gr MK) and 338 cal (250gr Lapua, 300gr MK) WILL make the trip. Would think the 208gr Amax and 210gr Berger will make the trip.
Hopefully, the guys from ABC bullets will be able to push their 408's and BMG bullets out that far.
The Amax 50cal has no issue going that far and can do so quite accurately.
Just remember that when shooting that far AND if you have enough up, initial muzzle velocity is not that big a deal. It's all going to slow down. Accuracy, consistency and little vertical dispersion is more important.
Hope this helps.
Jerry