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Question (inspired by this question, and of course the 1998 classic Armageddon)

Is there a conceivable reason that NASA or any equivalent space agency (ESA, Roscosmos, etc.) would need to send non-astronauts into space? It seems that for almost any mission, astronauts could be instructed on how to complete the necessary task beforehand or remotely. If not, machinery could also be operated remotely. Lastly, for planned missions (e.g. Mars), civilians with requisite credentials (e.g. Geology PhD) could be selected for astronaut training well ahead of time. In what situation would a non-astronaut of any expertise (or lack thereof) be required to personally accompany a mission.

Specifications

The time period to consider is near past to near future, implying that the only people trained to go to space are astronauts and the odd paying tourist. The technology level to consider spans the same time period (think shuttle era to ISS to Deep Space Gateway/Lunar/Martian missions. Any length of mission preparation up to the actual length of astronaut training (~2 years) is acceptable.

Criteria

You can invent any wild scenario to justify this (obviously "all the astronauts just got pink eye and the Hubble needs a bath" is a bit out of left field but hey go wild). If multiple valid scenarios are proposed, the most realistic/probable one will be the correct one. More weight will be given to the need to send non-astronauts than the want (e.g. sending scientists to do a sciencey thing rather than sending politicians for a P.R. opportunity).

Payload specialists were regularly flown by NASA until Columbia disaster. Payload specialists are not career astronauts and receive less rigorous training (not sure how less rigorous, but definitely shorter than 2 years).

The idea behind the specialists is that for a number of jobs it is easier to train a civilian to fly in space than to train astronaut to perform the job on the same level as that civilian specialist do. for example, if we want to take "fine art" photographs from space, we need to send out a civilian photographer rather than give career astronaut a top-level camera and tell him which buttons to push.

Alien's have encircled the Earth and demand to meet our leaders, or we meet our demise. They have read our communications so there is no hope in fooling them by sending the wrong people. Our only choice (maybe) is to send presidents/foreign diplomats/leaders into space with little time to train.

It's not the skill that's needed, it's the people themselves

Maybe the people have a rare genetic condition/blood group/other physical property that make them uniquely suited to the task at hand. Maybe a catastrophe is about to befall Earth, and it's determined that stashing some VIPs (high level politicians, important scientists, etc) on the ISS is the best chance to keep these people alive.

You need a lot of people

Maybe you're planning the colonization of, say, Mars, or more generally you need to (again) evacuate Earth. Now you're not going to get everyone out obviously, but you still want the most amount of people, and the most diverse group too. So either there will be more people needed than there are trained astronauts, or the diversity requirement will mean that most of the participants will be non-professionals.

Celebrity Big Brother IN SPACE!!.

In an effort to stir up public appreciation (and continued taxpayer dollars) NASA sends up a group of celebrities and then films their inept but charming hijinks aboard the space station! Celebrities in space turn out to be surprisingly competent, incontinent, sexy, grumpy, intelligent and profoundly annoying. It is the best TV ever.

The public cannot wait to send another rocketload up for the next season, and this time, the Chinese want in too!

Similar to pwaivers answer, but crazier (and absolutely NOT inspired by Space Jam) :)

Aliens have circled the Earth and are ready to conquer the planet. But they have to follow a particular chivalric code in invading planets, so they must give a chance to the populations they are about to invade.

They decide to organize a match of [insert your favourite sport here(1)], the team of humans against a team of invading aliens, on neutral ground. If the aliens win, mankind will be enslaved, if the human team wins, the aliens will leave the Earth.

The have set up a dome orbiting around the moon, where there is a gravity similar to Earth's and the match will be played, so we must send the best players of the chosen sport into space, to play a game that will decide the fate of mankind!

(1) if the alien players look like the xenomorphs, probably chess or bridge would be a safe bet

Budget Cuts

Space is expensive. NASA and other government equivalents cost a significant amount of taxpayer money to fund and NASA in particular after the cold war and the shuttle era are in a constant battle to publicise their relevance to the modern world and particularly to the Americans that fund it.

During the Cold War, this wasn't so much a problem because the government actually WANTED to have a thriving space program to demonstrate that they were equal to or better than the Eastern Bloc when it came to scientific advancements and in particular, space. But, the Cold War ended, and the shuttle era saw NASA shift to supporting a lot of industry in space as much as scientific research. Just take a look at some of the shuttle missions and you'll see that a number of them were for things like communication satellites, which meant that the shuttles had at least some commercial value.

Since then, I've noticed a subtle change in the articles in my NASA news subscription feed. They're about 1/3 Mars stories, 1/3 rest of the solar system, and 1/3 Earth imaging or climate change style articles. Why this latter group? Because the money has to be relevant to the people funding it in these times, and NASA has gone from a secretive cold war artefact designing Saturn V rockets to a public domain organisation justifying its existence to a taxpayer who is increasingly indifferent about Mars and the other planets.

I'm not saying any of this to push an agenda, simply to point out that in such circumstances it's conceivable that one government or another in the future will either severely cut or kill off completely the NASA budget. But, they still have things they want to do. How do you do things without a government budget?

You go commercial.

Privatisation has been proceeding apace in many countries for decades. Australia has been seeing it wholesale since the 1970s, for instance. Even NASA during the shuttle era were taking on commercial payloads and that may well continue without a shuttle fleet, but not on a scale that can pay for the science agenda. BUT, space tourism might.

So, you privatise NASA as a space tourism country with decades more experience than SpaceX or any other startup. You continue to run your space missions, but with a 50% trained crew and 50% passengers, who pay more than double the actual cost of their seat, funding the entire mission.

Sure, that means it's awkward getting those classified payloads in place with all the iPhones taking pictures of it out the windows, but then if you really cared about such things, you wouldn't privatise NASA in the first place, right?

Right?

Types of scenarios

Astronauts are selected to be the best and brightest. Many have military backgrounds, all have some type of science training (from their astronaut training if nothing else), all have university level degrees. They are also trained to operate in zero-G, which means that even if they are not the best at a given skill they may be better able to use that skill in a zero-G environment compared to a dirtside expert. For example, a specialist brain surgeon is doubtless better at brain surgery than a NASA generalist physician when operating on Earth, but the astronaut may be better at brain surgery in free fall simply because the specialist surgeon has no idea of how to operate in that environment. So any physical skillset is unlikely to make anyone a preferred choice compared to a trained astronaut.

Looking beyond physical skillsets to knowledge, even if no astronaut has expert knowledge in a particular subject, telepresence and remote operation allow for expertise to be provided from Earth without the physical presence of an expert. For example, if an alien ship with Egyptian symbols on the outside came tumbling past Earth, NASA would send up trained astronauts (including anyone who took archaeology as a minor subject in college) and have a battery of Egyptologists in mission control to translate the footage sent back.

Knowledge skills have some potential if telepresence is not possible. If a mission is required to operate under strict emissions control for an intelligence-gathering or special forces action then it might be that having a trained mimic who speaks colloquial Icelandic could be critical to infiltrating a space station to stop Dr Evil's nefarious scheme. Given the difficulty of being stealthy in space (and the unlikelihood of anyone allowing Dr Evil to build his space station in the first place), this is an extremely far-fetched scenario.

A slightly more plausible option is a "dying wish" type scenario. Billionaire CEO and astronaut John Smith has been critically injured in an accident on his company's newly built private space station in Earth orbit and will die in a few weeks. John Smith's injuries are such that he cannot survive returning to Earth. Jane Smith is sent up on the next resupply flight to be with her dying husband when he passes. While this is not strictly a must send an untrained person to orbit scenario, it is possibly plausible for near future.

Finally, going clean into fantasy realms, putting together Timst's answer and Thorne's comment about getting rid of the Kardashians - maybe it is the safest place to put someone. A person develops telepathic powers that they can't control and their radius of effect is increasing. They are innocent (and potentially very valuable when/if their powers are understood but it will soon be unsafe for anyone to be within a 10,000 km radius. Instead of killing them, stick them in a space station in geostationary orbit and do not route any manned missions within the danger radius.