About how many calories would it take to boil a gallon of fresh water?

A calorie (kcal) is the amount of energy it takes to raise the temperature of 1 kg water by 1 °C. Water weighs about 1000 kg per cubic meter, which is equivalent to approximately 3.79 kg per gallon. Assuming the water is at room temperature (25 °C), it will take about 284 calories to raise one gallon of water to near boiling (100 °C).

In order to actually boil, at least part of the water must be turned into steam. It takes 540 calories per 1 kg to turn water into steam, so it will take some additional energy beyond the 284 to actually bring the water to a boil.