> 7 days, most of which will be above the tree line.
> I want the absolute lightest cooking system possible, super ultra light.
> And I want to keep my expenses low, super ultra cheap.
> as little space as possible.
> Need to boil 2 cups of water daily for dinner is the goal.
> So I'm thinking alcohol stove.
If you really only need to boil 2 cups (474ml) of water per day, then you'll need about 20ml of alcohol per day, so 140ml in total. Call it 150ml, weighing 120g.
Then add a fuel bottle; a 250ml soda bottle weighing 20g.
For low cost, low weight and small pack size, go for a MYOG red bull burner (7g), and a MYOG cone clone (~25g, depending on pan size). If you go for a fissure cone clone, it will fit in the pan, in most cases (as will the burner, lighter, etc). Add a fuel measuring cup at 3g.
So that's 55g for the burner, windshield, pan support and fuel bottle. Just under 2oz.
Add whatever pan you've got, or want to buy, and choose a burner design to suit it; narrow flame ring/cone for a pot with a small base. If the burner doesn't suit the pan, you'll potentially waste a lot of fuel as the flame passes uselessly up the side of the pan.
Replace the lid of your pan with one made from a can wall opened up, flattened out, and folded over the edge of the pan.
Practice beforehand in equivalent conditions to convince yourself that you know how much fuel to take. And protect your stove from wind to make the best use of the fuel you've got; pack, rocks, etc. for a makeshift first stage windbreak.
If you were going really minimalist, you could forego the cone, and take three Ti pegs at about 2g each, but you'd have to make a very good makeshift windbreak, and you wouldn't get the increased fuel efficiency that the cone provides. Or just support the pan on handy rocks...
I've seen people suggest Esbit tablets, but my UK experience with hexamine blocks put me off solid fuel when I was a kid. I should have a play with Esbit, as I'm assured they burn quite cleanly, and have a higher energy density than alcohol. Of course, you can't so easily measure just enough Esbit to boil X ml of water...
Apologies for any 'egg-sucking' in this post...