The length of time that it will take to organize your Kilimanjaro climb will mostly depend upon how long it takes you to prepare everything and get the gear you need to climb Kilimanjaro. It also depends on your climbing experience and the amount of research you do before your trip.
Tour operators are often very quick to respond to requests and may even be able to organize your climb within a few days, depending on the availability of guides. Some climbers prefer to leave at least half a year or more to organize their trip and do the necessary research. Others are more spontaneous and only need a few days to arrange their tour.
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