Speaking of cheap wine, you get get 5L of ~14% red wine for about $9-10 here.
5L at 14% is 700mL of pure ethanol; that's 70mL ethanol per $1.
As far as I know, there is literally no cheaper way to drink in the US.
The cheapest liquor I've ever seen, which I did actually buy a bottle of, was 40% ethanol @ 750ML @ $5. (Distiller's Pride bourbon - it was actually totally okay bourbon, for $5, I'd buy more.)
That works out to 60mL ethanol per $1, pretty close, but still not as cheap as box wine.
TL;DR Buy the highest proof, cheapest box wine, and you'll get drunk cheaper than literally any other option on the market (that I know of.)