Leading to ~10,000 feet (or nearly 2-miles) of elevation change. From there, you can dig another mile underground, leading to 1-mile (under ground), or -1000 feet elevation, to a peak elevation of 14,000.
If a tower were built on the top of the mountain: you could gain another 2000 feet or so on top: so maybe 16,000 (a 2000 foot tower on top of the mountain peak) to -1000ft (1-mile deep from the bottom of the 4000-ft elevation valley), for a total differential of 17,000 feet.
Ignoring earthquakes and other issues, of course. :-) Just purely from a hypothetical perspective: working with nature and the natural landscape seems like it'd be better than "just" digging a hole.
EDIT: Repurposing abandoned mine shafts might be worthwhile, depending how deep they are.
Gravitational potential energy is (approximately) linear in height. I say approximately because this assumes constant g (which is a good assumption when h is small compared to the radius of the earth, which it is).
And in fact, a consequence of gravity's 1/r^2 nature is that one is only subject to gravitational acceleration from what is beneath them (shells above cancel out), so mine shafts are less efficient than towers (the effect size is small to the depths we can mine).
So adding more height doesn't help, and if that height is underground it could actually hurt net efficiency.
Efficiency in this context refers potential energy stored per unit height. The field is conservative no matter what you build.
If you picture a dense weight like a cannon ball on the end of a string you're right, but if you're digging down n meters, encasing n/2 meters worth of dirt and moving it up and down the free n/2 meters of shaft, the energy storage would indeed be proportional to n^2.
I don't know anything about the field and had the same reaction you did, but considering parent is running a startup in it they're either a lunatic that doesn't know the equivalent of FizzBuzz or there's something we missed on first inspection, and we should charitably assume the latter...
The single shaft vs multiple parallel approach does seem a bit risky in the early days. If there's a 10% failure rate, and you built one shaft, that's a 10% chance of an existential threat to the company. 10 shorter shafts mean one will likely be inoperable.
Of course over the long term worrying about this doesn't make sense. Once you've scaled, 1k large vs 10k small shafts would not matter from this perspective.
Best of luck mate!
There's another thread on this comment page talking about why height is important. Please see this slide in our presentation illustrating it. https://docs.google.com/presentation/d/17FI-jrI9RWS3q7Ng44Yh...