In reply to ClimberEd:
I'd just like to mention that for the Crochues traverse and the south west ridge of the Belvedere, although the climbing is no harder than French 3 and would present no problem for anyone capable of soloing (I do both routes on my own every now and then), a rope is pretty handy for the two abseils, one in the middle of the Crochues and one on the descent of the Belvedere. Both sections can be downclimbed if you REALLY want... but if the OP isn't bringing any kit, then both routes might be just an inch or two out of reach.
I think that quite an exciting day out that requires no ropes is the ridgeline starting above the Montagne de Peclerey, between Le Tour and Argentiere, that works its way towards the Col du Passon over two lumps called the Bec Rouges Inferieur and Supérieur. The ridge has various subsummits and cols that present themselves as optional turnaround points if the day starts to get a bit much, and there are a few interesting rocky sections scattered along the way, getting harder as you get closer to the Col du Passon. There is also a short section just before the Bec Rouge Sup. on a flat bit of pocket glacier.
As soon as you gain the ridge from the steep alpenrose meadows below, the views onto the Argentiere glacier and the Le Tour glacier are stunning, and you've got the Aiguilles Rouges in all their splendour behind you. Being a ridge route it's a day out that I often do on bad weather days because it's pretty tricky to get lost, but it's really worth going up on a bluebird day, because the views really are as good as anywhere else in the Chamonix valley.
You can start from Old Argentiere or Le Tour, if you make it all the way to the Bec Rouge Sup., I think it's around an 1800/1600m day, respectively.