Short answer: A lot. Long answer: It depends on a multitude of factors, including body weight, pack weight, elevation gain/loss, terrain, and more.
Here’s a quick round-up of a few online estimates, plus some useful tools, for gauging your total caloric burn on your next backpacking adventure:
Backpacker Magazine suggests a simple calorie estimate based on body weight and the general intensity of the day’s activity. For a strenuous day of backpacking with a “heavy” pack (no weight range specified), they suggest 25to 30 calories per pound of body weight. Using my 185-pound self as a proxy, that’s 4,625 to 5,550 calories.
The Mayo Clinic lists calories burned per hour for a variety of activities, including backpacking (though no pack weight is identified). According to the site, a 160-pound person burns 511 calories per hour and a 200-pound person burns 637 calories per hour. For eight hours of activity, that would be 4,088 to 5,096 calories, depending on weight.
For the criteria I used (185-pound person backpacking for eight hours with a moderate to heavy load), estimates range from roughly 4,600 calories to more than 6,300 calories.
Based on my personal experience, I would say that range is a reasonable approximation, with steeper and more strenuous terrain leaning toward the high end, and easy to moderate terrain toward the low end.