To be honest, as many days as you can get. In Chamonix, you might spend weeks and yet not see everything. However, even if you just had a few days, a vacation to Chamonix in the summer might be worthwhile.
Depending on the time of year, during the busy season, access to the most popular activities needs pre-booking. It might take longer to be in town.
A whole week would be ideal for those who aren’t gifted with infinite vacation time. You could visit the most remarkable sights in a week, go on some of the best day treks, eat delicious alpine foods, and yet have time to unwind.
The activities may be packed into even 4 or 5 days, though.
To visit the Aiguille du Midi, travel from Montenvers to the Mer de Glace, and do a few treks up and down the valley (Lac Blanc is the most popular), at least three days are needed.