What would be the biggest round you could safely rechamber in the action. with out reheat treating.

#1). What exactly is this rifle? A pattern of 1893 or a pattern of 1898? I am not sure when the '98s took over but ... there were some small ring (receiver ring) Mexican Mausers much sought for light rifles. 98 design.

#2). What year was the gun made? Proofed? Often if you take the wood off and look at the bottom of the receiver above the trigger with a magnifying glass you will find a proof mark with a date. Often. Not always.

#3). What do you mean by "biggest?" Length of cartridge? Bullet diameter? Pressures? The action was designed around the Mausers. 7x57. 8x57. 6.5x55 Swede for that matter. 7.65x53 Belgium. Any of these in original pressure loads should be "O.K."

#4). De Haas in his Bolt Actions book suggests that rifles made before 1930 be restricted to the older, lower pressure cartridges of that era. Nothing wrong with those cartridges. 7mm Mauser for starters. What is the gun now? 6.5 Swede is just short of .270 performance.

Of course you don't mention your intended target? Deer? Varmints?

8mm Mauser is another, larger. Actually .30/'06 is from these pressure levels in factory loads but a bit long, probably -- 63mm.

Varmints? .250 Savage. The original .257 Roberts is from this era. You would want to avoid the +P loads. .244 / 6mm Remington is also from the 7x57 case but loaded to more modern pressures.

De Haas suggests that quality Mausers only made after 1930 be used for more modern high pressure rounds like .243 or .308 or .260 or .270 or ???

Re heat is a great idea if you intend the newer rounds. And even if you don't, would not hurt. .35 Whelan goes back into this era as a wildcat. Todays ammo might be a bit warm. I believe there was a .375 Whelan... Still a wildcat. luck.