Good question, and OP wasn't a faggot and actually put round count in, because many firearms are immune to age with simple proper care. The 22lr bolt action is a good one, as other posters mentioned, barrel will last forever, tens of thousands of rounds, I've heard there can be a loss of accuracy at 200,000 shots, but does that mean loss of match accuracy or functional accuracy? Either way it sits easily at the top of round count consideration. With modern gun steels and good parts, it may not need repairs or major replacement of anything for an extremely long time.
As a technical note, even a high pressure round for rifle can be handloaded to lesser pressure, even very low pressure with lead bullets especially. If one keeps the pressure low enough, barrels may theoretically last, again, hundreds of thousands of rounds with careful light handloads. A Mauser with a claw extractor to help control headspace, or a rimmed or truly belted cartridge, will help these rifles maintain headspace even with a high round count with even high pressure standard rounds. The components are tough and will last just as long as a 22lr counterpart.
Break actions are good, solid guns, but wear and recoil can eventually cause the gun to "get off the face" and have potentially serious problems. A gunsmith with simple hand tools and knowledge can fix these, however, and keep them running if that's an option. They are tough and reliable, and a good consideration.
Single shot falling blocks like the Ruger 1 or Winchester 1885 are potentially surefire choices. They are often chambered in large cartridges well suited, if not originally intended for, blackpowder. They are the strongest action, very simple, not much to go wrong. How many rounds through my Uberti 1885 45-70 with lead bullets and low pressure loads would it take to wear it out, or destroy the barrel if no mercuric primers and/or blackpowder is used? Perhaps, again, hundreds of thousands, with a rimmed cartridge to help headspace to boot. This might be your answer.
Magazine fed levers have too many parts, too much to go wrong, in my opinion. At least many can be simply loaded and fired single shot if other parts break in many cases. Probably not the best choice.
One might get good service from a pump rifle, and many pump shotguns seem to have incredibly high round counts without major issues. The guns can get sloppier and looser, but they still work. Desert Punk's 1897 might be a bit more fragile, but modern guns like the Ithica design or the R870 or M500, when built right with good steel, might last you an unbeliavely long amount of rounds fired. Simple, stupid, with shotguns low pressure and no headspace or rifling to worry about, this is a good choice.
Automatics are always a questionable choice, parts tend to fail and break on them, critical things they need, sometimes parts that can't be made by hand or rigged. Plastic handguns will eventually fail due to age rot, if not this century then eventually. Some steel handgun automatic frames will crack with high round counts, both require certain parts at round counts to function reliably. Some assault/battle rifles MIGHT have things as bad as receiver failures, and they too eventually need parts for reliable function. Even if the gun in total isn't worn out, they are reliant on parts and repair at high round counts.
Revolvers might be another great choice, there are many high grade revolvers that have survived 100k+ or far more round counts. Its not the longevity of the gun in terms of rounds, in fact in handguns it might be king, the problem is these guns "get out of time" and need to be refit. If this fits OP's major work or repair consideration, this alone may keep the revolver from being a super great choice.