Someone earlier in this thread wrote he/she built a 13 pound bike for $500.

With all respect, I suggest he/she weigh the bike again.

I have light everything, including a ti spindle and alloy crank bearing cups.

My bike weighs considerably more than 13 pounds.

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In a conversation with the Mercian folks sometime back, they advised against trying to save weight in the frame, especially if one intended to ride on the street.

A famous builder, on his website, has an essay with a similar sentiment, in which he advises saving the weight in the components, especially the wheels, and not in the frame.

For a fixed gear frameset for the street, I favor Reynolds 853 steel (Reynolds 725 steel for the stays and fork), first, and ti a distant second (second because of the cost and the option of dollars better spent elsewhere).

I think the lightest day in and day out wheels one can put on a street fixie would have Dura Ace small flange hubs; an EAI alloy cog; 32 Sapim Aero CX-Ray spokes in back and 28 in front; Sapim Polyax-Nipples; DT Swiss RR 1.1 rims; Salsa ultra-light tubes; and, Continental Grand Prix 4 Season tires.

A person could spend a LOT MORE money and get a marginally lighter but much more fragile wheel than the one described immediately above.
For less money, and only slightly heavier, overall, one can get wheels with significantly lower inertia by choosing the Cane Creek Volos Track wheels: these wheels have no nipples in the rims.
I weigh 235 all up and have ridden the Cane Creek wheels for a year without doing them any favors.
They hold up to hard riding and feel weightless.

Titanium bolts, super-light stem and seatpost, super-light saddle, and a super-light front brake (or no brake) can add up in weight-savings and dollars.

I tried feather-light pedals and they didn't perform well at my weight.
They might for a lighter rider.
In the end, I opted for Shimano's most expensive and lightest two-sided Mountain Bike pedal, which weighs one and a half times and costs twice as much as the super-light pedal, but which takes a beating and keeps on working elegantly.

I put a relatively inexpensive crank on my bike because I liked its bolt pattern for reasons too complex for this post, and it would have cost three times as much to get a lighter crank.

So, my only concessions to weight involved the steel frame, Shimano pedals and good-enough crank.

Oh yes, and definitely go 3/32's and not 1/8.

Anyway, after spending $2400 on light parts, my bike doesn't come anywhere near 13 pounds.
I could not have made it any lighter by throwing more money at it, and I probably could have come in within a few ounces of my final weight for a third of the money.

I consider 15 pounds the practical limit, money-wise and reliability-wise, for a street fixie.

Start with a cheap steel frame and use it as a place to park parts as you acquire them over a few years of collecting, and then get a custom Reynolds 853 frame and transfer all the fancy parts to the custom frame.
Personally, though, I think having a cheap frame with decals helps camouflage the bike.
If I had it to do over, I would go with a Cane Creek headset in order to not have KING on my bike, beckoning to the over-curious.

Take two, three, four or five years to do it.
It makes for a nice hobby and costs a lot less than a car.
Gosh, it costs less than the gas and oil for a car.