In reply to _hs_:
With regards to admin in Nepal, they are all over it, you are not even going to get near the Khumbu without a national park permit as well as a TIMM permit ( Timms is a tracking system for trekkers, you are logged in and out at official checkpoints and supposedly at some point if you went in and did not come out you are classed as missing)
On a mountain any Nepali sadar or sherpa can request you show your permit to climb the peak, if they catch you without one they get a nice reward fee from the authorities, so yes there are lots of eyes to spot you.
Loads of peaks are not classified as requiring a permit mostly the ones on the Everest base camp trek or close by. Up above Gokyo there are some crackers that open without permits.
You have to understand that the officials do not want western tourists to get killed as it is very bad for business. To them it is inconceivable that anyone without a sherpa can climb the higher peaks.
I did a 4 week self supported trek in the Khumbu in 2012 and had a great time not doing any peak above 5600 meters. Once you start looking at 6000 meter peaks you are going to have to scale up your kit a lot. There may be crevasse, technical climbing sections, all on your own.
Have you been to 6000 meters before, no matter how fit you are you are going to find it a whole different world up there, I am only a punter and been to 6500 meters and 6000 meters on a handful of peaks very hard to convey to someone who has not how hard it all is.
My advice is A. go to Nepal it's amazing, go into a agency and talk with them what you want to do, they will fit you up with what you need if you are adamant you want to go above 6000 meters.
B go and do your own thing and stay away from the higher peaks.