TL;DR: 12-15km Earth-equivalent altitude, because you start needing spacesuits and pressurized habitats instead of a nice warm coat and a tent.
You might be OK if you could launch rockets to drop pressurized accomodation for you at high altitude, but you might consider that cheating.
- You need a low-gravity world to keep very high mountains.
- Mountains on low-gravity worlds are easier to climb than mountains on high-gravity worlds (cos you can carry more, and jump higher)
- Low-gravity worlds have a larger scale height so the air pressure drop (and hence oxygen partial pressure drop) is lower for the same altitude gain on a high-gravity world.
There's wiggle room in all those statements though, so they aren't strong arguments alone.
Okay, so you carry supplemental oxygen, right? But that runs you into a different problem: there is a limit to how much you can carry
The supplemental oxygen used by mountaineers is inhaled via an open-circuit respirator. This means that any oxygen that is not taken up by the body is exhaled into the atmosphere and effectively wasted. This means that they end up carrying a lot of oxygen but rather less of it is actually used in metabolism than you might initially expect.
There is a technological solution to this in the form of rebreathers. These are more commonly associated with diving, because they let users spend more time at depth without having to carry huge amounts of additional air. As a very brief summary: rebreathers scrub excess CO2 from exhaled air, and keep the oxygen partial pressure constant by trickling in oxygen from a tank as needed. This ensures that all the oxygen in the tank can be consumed by the user.
Using rebreathers for high altitude mountaineering isn't a new idea: The Use of Closed-Circuit Oxygen in the Himalayas
Two days before the first ascent of Mt. Everest in 1953, Tom Bourdillon and Charles Evans climbed to within 90 m of the summit at unprecedented speeds. By breathing pure oxygen from a closed circuit, the pair were able to obtain an enormous physiological advantage. Unfortunately, due to a malfunction in Evans's circuit, the pair abandoned their attempt on the South Summit. For many who used the circuit in the 1930s and 1950s, the device proved too heavy, uncomfortable, and tiring for mountaineering.
You will not be surprised to learn that rebreather technology has advanced considerably since 1950, and rebreathers used in low pressure air (rather than at high pressure under water) are a little simpler and safer than their conventional underwater cousins. A modern mountaineering rebreather would be much lighter and smaller and more reliable than their primitive 50s forebears, and would make oxygen supplies go much further. There would still be technological hurdles to overcome (because operating rebreathers at very low temperatures is probably not something that's done very often) but there's no reason they should be insurmountable.
(note that rebreathers form a part of the primary life support system of modern spacesuits, for an example of their use at low pressures)
You're still subject to exponential losses, but when the exponential term is small enough then your maximum altitude increases considerably. The practical gain from a rebreather might be a tenfold reduction in O2 consumption.
It would be tempting to think okay, you need to cheat, bring in a helicopter. But helicopters have altitude limits, too. I don't think they can actually go much higher than mountaineers.
This is true, but given your example of "Elon Musk" budget you can consider the use of a small rocket driven lander, assuming suitable sites can found to put it down. Maybe the thing could be dropped from a plane and glide to its destination using a parafoil, and use rockets at the end of the flight to fine tune its landing zone. It'll still be vulnerable to "weather windows", and making it robust enough to keep its cargo safe and in place during bad weather might be a problem too, but again: not necessarily an insurmountable one. And it is something that can be tested ahead of time and validated before any climb.
Would anyone ever climb, say, a 20 km mountain?
At a certain point, external pressure becomes low enough that simply being exposed to it is unhealthy even if you have a good supply of oxygen. On Earth, high-altitude pilots wear pressure suits:
The physiological-deficient zone extends from 3,600 m (12,000 ft) to about 15,000 m (50,000 ft). There is an increased risk of problems such as hypoxia, trapped-gas dysbarism (where gas trapped in the body expands), and evolved-gas dysbarism (where dissolved gases such as nitrogen may form in the tissues, i.e. decompression sickness). Above approximately 10,000 m (33,000 ft) oxygen-rich breathing mixture is required to approximate the oxygen available in the lower atmosphere, while above 12,000 m (40,000 ft) oxygen must be under positive pressure. Above 15,000 m (49,000 ft), respiration is not possible because the pressure at which the lungs excrete carbon dioxide (approximately 87 mmHg) exceeds outside air pressure. Above 19,000 m (62,000 ft), also known as the Armstrong limit, fluids in the throat and lungs will boil away.
That last one is worth a deeper look:
The Armstrong limit or Armstrong's line is a measure of altitude above which atmospheric pressure is sufficiently low that water boils at the normal temperature of the human body
This means that at some point between the altitude of the top of Everest and 15000m your climbers start needing spacesuits to provide counterpressure simply to let them breathe at all. At 20km they need a spacesuit to stop exposed fluids boiling away, which is likely to be an unpleasant way to die (unless their air supply was damaged in which case you just pass out in a few seconds).
More designs for mechanical counterpressure suits for use on Mars would probably do an excellent job, though I don't relish the idea of carrying a complete spacesuit up to the top of Everest and then having to put it on and climb another Everest and a half!
The need for a spacesuit, and a positive pressure respirator suggests that 12km altitude on Earth (or an equivalent atmospheric pressure on another planet with different gravity) might be as high as you'd reasonably want to climb because of the problem of eating and sleeping safely. You can have food-tubes in a space suit or pressure suit, and a clever vent system could allow you to pee and poop too. Damage to the suit might be very serious, and injuries are effectively untreatable unless you were able to bring a pressurized habitat with you (like a cross between a Gamow bag, a TransHab inflateable space station and a portaledge). This probably means that even if human mountaineering activity above 12-15km is possible, the fact that small accidents or kit failures can kill you in a quick and very expensive fashion probably limits climbing at that altitude (contrast with climbing 8000m peaks, where accidents kill you in a few hours in a merely expensive way, and sometime people get rescued).
edit: Here's a big problem: time.
Climbing is hard work, especially at high altitude, even with supplemental oxygen. It takes 11 hours to climb Everest from a base camp using supplemental oxygen, which is a rise of <3500m. The round-trip time to base camp was 18 hours. For higher peaks, you'll simply run out of time and energy and you're going to need to camp high on the mountain. No-one camps in the "death zone" at 8000m+ because the chances of you surviving your nap are slim. At 12000m peak basically demands a camp unless the climbers are superhuman. Napping in a pressure suit might work, and your oxygen demands are reduced, but those long and unproductive sleeps will cost you dearly in terms of the amount of food and oxygen you have to carry.
Permanent and pressurized high-camps would probably be needed... imagine the effort required to construct a sealed building at the summit of Everest and maintain life support systems up there. Imagine the fun you'd have providing power, given the low levels of oxygen to run combustion engines or fuel cells (and the effort required to refuel). Wind turbines and solar cells might have problems with the weather, to say the least, but might be the only practical options. This may also be considered cheating, but it is hard to see what you might do instead.