If you are planning a backpacking trip that includes hiking to the top of Half Dome, make sure that you specify that when filling out your wilderness permit application. You are required to carry a wilderness permit, and a Half Dome permit. The National Park now allocates 75 permits a day for just backpackers on Half Dome. Of those permits, 50 are able to be reserved and 25 are available on a first-come, first-serve basis. The process to obtain a first-come, first-server permit for Half Dome is the same as the process for a wilderness permit—visit a permit issuing station beginning at 11 the morning before your hike starts.
If you are hiking the John Muir Trail an are exiting over Donahue Pass, you must obtain a exit quota permit. Yosemite implemented this to protect the access for day-hikers, and to provide a quality JMT experience. The exit quota is 45 permits each day for backpackers finishing the trail over Donahue Pass. It is best to have flexible dates, and you can apply online for a range of dates on the JMT rolling lottery application.