We require every traveller to carry travel insurance, and we ask for the policy number before departure. This is not a formality — you will be several hours from a hospital for much of the trip, and an air evacuation without cover is ruinously expensive.
Make sure the policy covers emergency medical treatment and repatriation, evacuation by air, trip cancellation and curtailment, baggage loss and delay, and any activities you have booked. The detail people miss is altitude: standard policies frequently exclude trekking above 3,000 metres, which rules out Kilimanjaro and Mount Meru unless you buy a specific extension. Balloon flights are sometimes excluded too.
Buy it when you pay your deposit, not before you fly. Cancellation cover only protects what has already been paid, so a policy taken out in the last week is protecting almost nothing.