It is theoretically possible to hike very cheaply. I met a hiker - Cody Coyote - on the AT in 2013 who was basically a transient kid who had been hiking up & down the AT for several years. His food was basically what he could scrounge out of hiker boxes & other handouts. I also met the afore-mentioned Grandpa who was also hiking on the cheap (in blue jeans, no less).
I'd recommend budgeting more money rather than less. I'd dare say that no one ever said that they couldn't complete the AT due to having too much money. I've seen people have to bail due to running out of money.
To also minimize expenses, start later (assuming NoBo thru; less cold-weather gear = lighter pack), hiker longer hours/miles, and finish the hike quicker. Less time on the trail = less money spent.
Good luck on your hike.