I'd guess that the absolute cheapest method would be "Cowboy coffee". You add the ground coffee and water to a pot and heat it up with whatever heating apparatus you have. Remove from the heat just before boiling and stir. Letting the pot sit for a few minutes should allow most of the grounds to settle to the bottom and then pour off the liquid from the top. And to be a real cowboy, don't fret if it does boil a little -- it saves you the chore of stirring.

If instead you heat the water first and then stir in your ground coffee, the result ought to be close to the result of James Hoffmann's french press technique (where you give the grounds time to settle, just like the cowboys did).

Single vessel. Single heat source. Cheap. Works with preground coffee, to save the expense of a grinder.

I'd guess that the absolute cheapest method would be "Cowboy coffee". You add the ground coffee and water to a pot and heat it up with whatever heating apparatus you have. Remove from the heat just before boiling and stir. And to be a real cowboy, don't fret if it does boil a little -- it saves you the chore of stirring. Letting the pot sit for a few minutes should allow most of the grounds to settle to the bottom and then pour off the liquid from the top.

If instead you heat the water first and then stir in your ground coffee, the result ought to be close to the result of James Hoffmann's french press technique (where you give the grounds time to settle, just like the cowboys did).

Single vessel. Single heat source. Cheap. Works with preground coffee, to save the expense of a grinder.