What's the least amount?

you could go bare bottom and just use floating plants, or plants like java fern anubias and moss tried to wood.

I think it depends on the plants you want to keep. Some plants could stay down and spread roots with a very shallow depth, others like swords, dwarf sag that kind of thing are going to need at least an inch.

I generally use about 2.5 inches in my tanks and can keep everything happy.

Is there a specific reason to want to not use substrate or try and use less?

Bare bottom with glass or clay pots filled with substrate and plant the plants in those.

I used a big bag for all my tanks and the last tank only has around 1" and maybe 2" in on one side of the tank.