Just booked myself 12 days in the Alps, arriving in Chamonix. Never been before, not into too much detailed planning but I figured the Tour de Mont Blanc would be a good enough thing to do for a first trip. A quick look at the route seems to suggest that 12 days is more than enough time with long days and surreptitious bivvying wherever I end up so does anyone have any tips for side trips? Some slightly higher summits that could be reached without climbing or crampons maybe? Or variations that avoid some of the crowds?

I arrived back from doing the TMB last week so hopefully I can offer some help...

Firstly I'd recommend buying the Cicerone guide by Kev Reynolds. At £13 it's pretty expensive for a pocket guide but is useful and interesting to have.
We started from Les Houches (south of Chamonix and on the bus route from Geneva airport) and took 11 days doing the tour anti-clockwise. We found this plenty of time, especially when wildcamping as you can walk long days to cover more distance than the book suggests. Reasonably we could have done it in 8-9 days had we not taken half a day off in Courmayeur and an afternoon and a long morning off in Argentiere.

Detours can be taken in several places, many good variants are described in the Cicerone book. From Col de la Croix du Bonhomme (day 2 for us) we detoured to Col de Fours and Tete Nord des Fours (2756m) which offers spectacular views. This also opens up wildcamping options off the path high above La Ville des Glaciers. We actually camped by the farm buildings in the village which enabled us to make use of the good, clean public toilets!

In the Italian Val Ferret there is the opportunity to stay quite high from Refuge Bertone to Bonatti by visiting the Tete de la Tronche (2584m) and Tete entre Deux Sauts (2729m). This was a particularly good place from which to view the massif as it's so close to the path at this point and we barely saw anyone all day as most people skirt around the side of these mountains.

. Tete de Ferret (2716m) on the Italian/Swiss border isn't that interesting but keeps you high up for longer too, without need for crampons.

I'd also recommend the variant from Champex up the Val d'Arpette to the Fenetre d'Arpette (2665m). While not a really high walk, the TMB offers plenty of good scenery and for the most part I didn't mind not being on the summits. I'd love to return to the area, prepared for some proper alpinism, but if you don't want to carry crampons and axe the TMB is pretty good.

Some good camping spots we found:
There are two designated camping areas (free of charge) between Chalet de Nant Borrant and Chalet de la Balme, south of Les Contamines.
By the farm buildings at La Ville des Glaciers (apparently wildcamping isn't allowed but some other guys were there and the farmers didnt stop us- not all that wild as it's just off the path but you get great scenery)
Near the Alp Superior de L'arpe Ville, above lac du Miage.
In the valleys above Refuge Bonatti
In the woods just north of La Fouly - we paid €10 each to stay at the campsite in La Fouly, which was nice, but as we started walking the next day we realised that there were good spots just 15-20mins along the path. Also note, the only place to get cash between Courmayer and Argentiere that we could find was by getting cashback from the shop in La Fouly. If you spend more than 25CHF you can pay by card in the shops in La Fouly and Champex, which worked out fine as there were 4 of us, but if you're going alone you probably wont spend this much- think about getting Swiss Francs somewhere before you get to Switzerland!
High in the Val d'arpette - about 2200m. With a bivi it'd be fine, we struggled to find enough flat grass for tents here but did eventually succeed.
When you get back round near Chamonix, wild camping is a bit more difficult as it's a busy area with lots of ski slopes. We wished we had bivvies at this point but managed to camp above La Flegere without trouble.
If you take a tent and fancy some luxury, the campsite at les Frasserands near Argentiere (called Camping/Chalet Pierre Semard) was really good and only €5 per night. We stopped there to stock up on food (good supermarket in Argentiere), plus there's a bar at the site.

Hope that helps a bit, let me know if you have any other questions and I'll try to let you know what I found.

Many thanks for the detailed replies. Good information.

I also did the variant suggested above and wildcamped not far from Ref. Les Grands. The refuge isn't marked on the 1:50k map I have but from the description above it's fairly easy to find.

Just flicked through the guidebook in Borders. One easily viable alternative which didn't seemed to be mentioned in there (maybe I missed it though) which is obvious from the map is the route via Lacs Cornu & Noirs further on from Le Brevant up high in the Aguilles Rouge ridge. I used my extra days to do that as part of the start of a "second lap" and then back down the other side of the Chamonix Valley where close up views of the Glacier Argentiere had the biggest wow factor of the whole trip for me... no "Ice Grotto" with a slobbery St Bernard to have your photo taken with like at the Mer de Glace but much more awe inspiring.