Thin to win baby.
I’ve been wearing the Wigwam ironman pro quarter socks for years
http://www.runningwarehouse.com/Wigwam_Ironman_Triathlete_Pro_Quarter_Socks/descpage-WITPQS.html
jaretj
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I have some Balega socks that are super thin and very nice. Not wool though in case you like wool.
These are thin and good.
Defeet are my all time favorite.
DryMax Hot Weather socks are my favorite. Very very thin except for the small areas on the pressure points of the feet. Even as thin as they are, they hold up excellent. Expensive but worth it ($13/pr on runningwarehouse.com).
Swiftwick Pulse.
Thanks all. Good suggestions.
If you can find some of these Nike Golf socks, grab 'em. Super-thin and wear incredibly well. Some of the best socks I have ever owned.
I got my last 3 pairs off eBay as I think Nike has discontinued this style.
I have been wearing Darn Tough ultra thin running socks. They are rather thin, stay dry, no blisters, no hot spots. And they really are “darn tough”.
The injinji liners are pretty darn thin.
Foot powder and no sock.
Vince
www.tribikerun.com
DryMax has a hyper thin that is their thinnest sock.
Balega Hidden Dry Socks are the thinnest I’ve found.
Another vote for Balega - although they wear out so fast!
Target. Women’s polyester, no-show socks. Fit my men’s size 9 feet perfectly. Don’t hold water. Dont bunch up. Awesome color selection. $1.50/pair. Occasionally they go on sale for $0.99. 
Might not be the thinnest, but I love my Switftwicks! The “0” height would keep weight pretty minimal. I found the blend of materials and weave easier to put on in transition. The Vibes were a little thinner than the Aspires, but don’t have much compression… if that matters.
What’s your goal? Just curious?
I’d say Swiftwick Pulse is by far thinnest sock I’ve ever used. I prefer the Aspire overall for racing, but the Pulse is super, duper thin. Another 2-3 clicks on the BOA on my bike shoes vs the Aspire.