You didn't mention budget, class of rifle, or if you handload. If you like autoloaders you can get .50 Beowulf cal uppers to fit AR-15's. This carridge is largely a handloading proposition, though.
A .338 win mag is a safe choice, and useful for deer, as well. Ditto the .35 Whelan. The .375 H&H has been hammering game for 90 years, and is a grand old cartridge. All three of these calibers are powerful, practical, and available in medium priced rifles. Factory ammo is pretty easy to come by in the 338 & .375, a bit harder in the .35 Whelan.
Shop around, you can probably find some used .340, .378, or .460 Wheatherbys under $1000. These are stout guns to handle, whether for recoil, or for cost of shooting.
I have a Ruger #1 like Gewher's, except in .375 H&H, and it is nice shooting rifle. A bit hard to hold with full power loads, but extremely accurate with lighter handloads. You can also find them in .458 Winchester, .416 Rigby, .416 Remington, and .405 Winchester. The latter is a handloading proposition. Browning also has a cool .45-70 single shot, but expensive.
In lever guns, there are medium priced .348 Winchesters, .375 Winchesters, .444 Marlins, and .45-70 Marlins. It's pretty easy to find imported Sharps single shot replicas under $1000 - most are in .45-70, some in other calibers.
Lastly, a couple of eccentrics - the .358 Winchester, which is a .308 case necked up to .35 cal - available, but not common, in a variety of rifles. Not much kick, but it still makes big holes. Remington also now has out a re-issue of its .350 Remington magnum, which is more of the same, basically.
Thought this was a simple question, huh?