Himalayas

@anthony: I can recommend Nepal out of all the Himalayan countries. Not only is it affordable and easy to travel there, but the locals are welcoming and friendly. It will be more than just a mountain experience. Its also unique in that you can do independent "tea house" trekking, which is the most affordable, where you won't need to worry about food and tents etc (which otherwise porters would have to carry for you). Costs vary considerably depending on the number of days, permits, camping vs staying in lodges, whether you use a porter, guide or a "porter-guide", whether you fly locally (like Kathmandu to Lukla), and of course which agency you decide to do it with. If you care to narrow some options down, I can get a quote for you if you want.

Bear in mind that the Himalayas do not readily offer solitude along the regular routes. The Annapurna and Everest areas are packed in peak season. You'll have to contend with locals, porters, yaks, goats, donkeys and other trekkers on the trail. Naturally you can get off the beaten track a little and there are less popular, quieter regions.

Normally the passes do not go over 5600m.

I would recommend a minimum of 10-12 days trekking, adding 2-3 days in Kathmandu on either end. There is lots you can add to that if you have time (which is highly recommended). The peak seasons are October-November and March-April, though May (pre-monsoon) and December (early winter) are also good, with less people. You can actually trek all year round, you'll just have to contend with more rain and clouds, or more cold and snow (which closes some of the high passes).

Yes you still get a mountain experience even in the lodges. Take Gorak Shep for example, which is the last set of lodges before Everest Base Camp - it lies at 5100m with Pumori towering overhead. Then there are ones like Chukkung and Langden (Everest Region) which are in quieter valleys, and around Tilicho Lake at Annapurna. You can also do lots of nice day trips from bases like Gokyo, though the lodges can get full there. Endless choices, really.

You'll have to choose an area along with the length of time, which given the above minimum time frame, you could choose from among Everest, Annapurna, Langtang / Helambu, Upper Mustang (which may not fit the budget cause of the permit costs), Lower Dolpo and even the remote and forgotten Humla and Jumla regions. Again, did I mention endless choices?

High permit costs have more to do with certain (politically) sensitive areas along the Tibetan border, which has now turned into a revenue earning opportunity.

Oh, forgot to mention: a big factor is whether you want a porter and/or guide, and if you want a complete package which includes Kathmandu sightseeing and accommodation. If you are travel-savvy enough you can certainly do it all independently.