In reply to Aled Williams:
You may find this article useful, section on acclimatisation near the end:
https://www.ukclimbing.com/articles/destinations/how_to_climb_mont_blanc_-_...
People do acclimatise at different rates. Bald Eagle's suggestion is a great one. You could walk up to the Albert Premier Hut and do some afternoon glacier refresher. Perhaps do the Tete Blanche and Petite Fourche the next day and stay up in the hut for the Tour on the following day.
Staying at the Torino hut for some of the short easy routes is also a great suggestion. All sorts of options on the Toule and the two Flambeau, possibly the Marbrees. Cosmiques Hut is also great, assuming you are comfortable on the narrow exposed snow ridge going down from the Midi. Arete Laurence and a route on the Lachenal or Gros Rognon are easily accessible. Mont Blanc de Tacul is another option from there. It is 4,000 metres plus but conditions can be very variable - check with the OHM in town for the latest updates. Wind slab, needing a ladder across bergschrunds and teetering seracs can make this a serious route out of all proportion to its grade and so it may be best to do the Gran Paradiso suggested below.
Doing a 4,000 metre peak such as the Gran Paradiso should then set you up nicely. If the weather is poor in Chamonix or for the Gran Paradiso Zermatt is a good ( but expensive ) place to acclimatise. Including one of the shortest and easiest 4,000 metre peaks, the Breithorn.
Obviously, we are all assuming you have appropriate knowledge, skills, equipment. Good luck.