The largest expense is transportation, particularly when armored units are used. Ground combat units are moved by rail and ship, which is actually quite expensive, mainly because an armored division has over five thousand vehicles and thousands of tons of other equipment. Railroad cars have to be rented and ships leased for these movements, which take over a month to complete. Even when military transport aircraft are used to ship material more quickly, there is additional expense for fuel, spare parts, replacement equipment, maintenance and crew pay. Military aircraft are purchased with the idea that they will last for decades, flying only a certain (relatively low) hours a year. But it is also realized that there may be a war and the aircraft must be able to fly a lot more hours in a year. Same with all military equipment. For active duty troops, they don't get paid much more (up to a few hundred dollars a month per man for combat pay), but many simply use their equipment more. When troops are sent overseas, more money is spent on tents and other temporary shelter. Food expenses go up because more expensive field rations (MREs and the like) are eaten. American troops also expect more amenities; everything from email access to gyms.
Estimates are made, from past experience, of how much equipment will be lost to accidents, and money is set aside to buy replacement equipment each year. Estimating wartime losses is more difficult, mainly because there is far less time spent at war than in peacetime operations. The trend over the last two decades has been for fewer and fewer wartime casualties. A war against a more evenly matched foe would produce more casualties, but fewer than in the past because troops are now better armed, equipped and trained.
Because of their low casualties, these last three wars were called, with some justification, large scale training exercises. And there was a significant training element to all this. During these three wars, there was a foe, no matter how diminished or inept, who was shooting back. This makes our troops "combat experienced" and far more effective in battle than troops who have not trained a lot and been shot at in battle. This is an enormous advantage, and this is what all the extra costs are buying. Rarely is this mentioned, but it's the key element in keeping the friendly losses down and the enemy flat on the ground.
Calculating the cost of a war, especially smaller wars that use professional troops who will return to their peacetime eventually, is largely a political exercise. These calculations also tend to ignore the valuable training and combat experience gained. These wars and peacekeeping exercises are, in fact, extensions of the usual training routine. For most of the troops involved in our last three wars (Persian Gulf 1991, Kosovo 1999 and Afghanistan 2001), the major additional expense was fuel, spare parts, ammunition, food and amenities, additional pay, medical care and transportation.