So your irrigation system is free, timers are free, water is free, sisal or other twine is free, rhizomes are free, cable/pulley/hangers are free, posts are free, your drying racks/system is free, and your fertilizer is free.
I have that all correct then? Did I miss anything else? Domestic pellet hops are around 75-85 cents per ounce. You think you spend less than $10 per pound including time and opportunity cost to do it?
That would be quite the feat. As I previously noted, though, if you enjoy it and you do it as a hobby and discount all of your time and opportunity cost, then you might be about "even" to produce hops you don't even know what the alpha acid content is and get minimal harvest until at least year 3.
Unless you are going to itemize the costs for your set-up and prove it, let's just say I have my doubts (because I have done the analysis for my setup which is not terrible expensive on its own).
Set up a hose with a bunch of holes in it. Use an old hose nobody wants, get a used one or buy a new one for $10. You can find used timers and whatnot for next to nothing (do a craiglist search...a ton of results come up) if you really want to do that, but I don't see why watering your plants should require so much technology.
During the rainy season here, the water is free. Otherwise the cost is minimal...$1/month or less to turn your hose on for a few minutes a day.
It doesn't cost me anything to go outside and turn the hose on for a few minutes. Yes, it's a hobby, but besides that many people have lawns, plants or go outside to jerk around once a day. No, I'm not counting that time...that's just silly.
Rhizomes are $5-$10...one time cost. You will get POUNDS of hops for them after a couple of years growth. You could easily get a pound the first year if you get a good rhizome that does well in your environment, properly fertilize it and water it. Hops are weeds, they grow like crazy.
Twine costs next to nothing. You can get a lifetimes worth (not just for hops) for $20 or get a friend to give you 20 feet or so.
I've got them hooked up to the top of the house...no cost there. No stakes or pulleys.
Even if I did spend money on all the supposedly required nonsense, every cost is up front. To maintain your hop garden costs next to nothing. Just like upgrading to an all-grain system, all the cost is up front...you do save money over time if you do it right.
You don't need to know the alpha acid of your hops. You should not use them for bittering. If you wanted, you could send them to a lab to be analyzed, but this would waste a great deal of your hops and is very expensive.
If you're that concerned about alpha acid and you absolutely want to save costs, then buy some warrior, magnum or other high-alpha variety and make only low-hop, low-alcohol beers via all-grain.
I could tell you that you could get all your equipment for next to nothing used or converted from other materials, but apparently you consider (minimal) time to be so important in a HOBBY that it is far too cost-prohibitive.
It is true that this works better in certain areas for certain people. If you don't own your home, you may not live there long enough to get a good yield. You won't get a good yield in a small amount of soil. You will have to water the hell out of it in a dry climate. Et cetera.
In the end, if it works in your environment, it is well worth the absolutely minimal cost.
I've spent probably $100 TOTAL on my hops. This includes water (I water them directly during the summer) at ~$36, wire at ~$10, rhizomes at ~$30 and fertilizer at ~$20.
I built racks for drying them from leftover wood and some wire mesh I found. Took me less than 30 minutes...a one time "cost".
By all means, count your time if it's precious...but you better well count all the time you read about different varieties when creating recipes, the time it took you to go to the store and pick them out, etc. if you are going to count everything. At that rate, the least cost-prohibitive way to brew is to mail-order a bunch of pre-made kits.
5 minutes on the internet. 5 minutes to receive and unwrap. Done. Cheapest way to brew.