Short answer: more days than you think, and the extra one is the cheapest thing you will ever buy on this mountain.
Why the number of days decides the summit
Kilimanjaro isn't technical. There's no rope, no ice axe, nothing you need a course for. What there is, is altitude. You start somewhere between about 1,600 and 2,300 metres depending on the gate, and Uhuru Peak is at 5,895. Your body can handle that, but only if you give it time to catch up. Give it the time and most reasonably fit people walk to the top. Rush it and fitness won't save you. Marathon runners turn around at 5,000 metres while someone twice their age walks past them, and that happens every week up there.
Five days
It gets sold because it's cheap, and it's the short version of Marangu. You sleep at roughly 2,700 metres, then 3,700, then 4,700, and you go for the summit on the third night. That's a lot of height in very little time, and the number of people turning back shows it. If five days is all your budget stretches to, we'd honestly rather you waited and came back for seven than spent the money on a climb you probably won't finish.
Six days
The usual Machame or Rongai itinerary, and a genuine climb. Six days gives you one real acclimatisation day, the kind where you walk up high and come back down to sleep lower. Plenty of people summit on six. It's a fair choice if you've been at altitude before and you know how your body takes it.
Seven days
If you asked us to pick one, this is it. Seven-day Lemosho or seven-day Machame gives you two extra nights between 3,500 and 4,000 metres, which is exactly where acclimatisation is won. You reach the last camp before summit night less worn down, you've slept better, and you've had a day to find out how you handle thin air while you're still low enough to do something about it.
Eight days and the Northern Circuit
Eight days is seven days with the margin widened. The Northern Circuit, at eight or nine, swings around the quiet northern slopes and has the most gradual profile on the mountain. It costs more, because every day up there has a crew and a park fee attached to it. What you get is the best acclimatisation available and a path with almost nobody on it.
What extra days can't do
They can't make summit night easy, and they can't rule out altitude sickness. Some people react badly however slowly they go, and the only treatment that reliably works is going down. Days improve your odds. Nothing guarantees them.
Count your trip honestly
A seven-day climb means seven days on the mountain. Add the day you fly in, a briefing and kit-check day before the gate, and at least one day afterwards when all you'll want to do is sleep and eat. That's ten days in Tanzania for a seven-day climb, before you think about adding a safari or Zanzibar at the end.