In reply to Matt_Ivory:
Bit of an open Q
Cheapest - hitch hike and camp (wild camp where legal).
Eastern Europe will give you more for your euro - we enjoyed Slovenia this year - like Italian Dolomites, better Englsih speaking
Backpacking on a recognised route can be cheap - thousands of options for this. Hut to hut will not be cheap but rucksacks will be light and you'll get to meet lots of other people.
I've got a bias for the Maurienne, it is a lovely, unspoilt part of France and cheap (for France). camping is about 10-12 euros for a pitch per night.
Tour de Mont Blanc is popular but really busy and the infrastructure tneds to get more expensive/needs booking well in advance.