Let's say for the 2016 Federal Budget, someone makes an "accidental" cut and paste, and for that year and that year only, NASA's budget is switched with the military's. The military is ensured to have enough reserve funds to make it through the year, although some programs will need to be put on hold and numerous departments of the military will need to go into a standby mode. After a congessional and media uproar over how such a stupid switch-up made it past the double-checkers who "overlooked" the change in the budget, Congress reluctantly allows NASA $600 Billion in a once in a lifetime anomaly.
NASA is overjoyed, space enthusiasts are tickled pink, and all your average global people want to see what they'll do with this money.
So NASA gets ambitious. They aren't going to use all the $600 Billion in one year, but they'll basically double or triple their annual budget, pursuing things that are crazy and also down to earth.
First comes all those telescopes, probes and satellites. The 10-20 years it would have taken to launch them is now closer to 1 or 2 years. Most of them launch successfully, some don't survive, but it seems like almost every few weeks another groundbreaking discovery is made either here on Earth or in the cosmos.
Then come the rockets. Orion and SLS are now fully funded with plenty of money to spare, and need places to go. Venus, some Near Earth Asteroids, the Moon, Mars, Mars' moons, and large asteroids in the belt are immediately considered for destinations. Jupiter and its moons may have to wait for better radiation shielding. Orion becomes a global inspiration, and kids watch in amazement for years as Orion is paired up with interplanetary habitats that are taken further and further away from Earth to places that have never been touched by living creatures.
Maybe a few geniuses at NASA realize the money won't last forever, and decide to helm some projects meant for the private space industry, such as space stations, beyond LEO rockets, reusable spaceplanes, asteroid mining, and lunar research bases, so that when this $600 Billion begins to run dry, they can hitch some rides with private groups like ULA, Orbital Sciences and SpaceX.
Then comes extremely ambitious sci-fi tier stuff. Investigations into suspended animation, dark matter, warp fields, wormholes, antimatter, FTL travel, Interstellar probes, and long-term plans for manned exploration of the outer planets and Interstellar travel ramp up massively, possibly producing groundbreaking discoveries and inventions that change our entire way of living and thinking. Robotics also gets a leg up, and robots alongside astronauts are extremely common.
Then come your less obvious benefits. NASA needs a lot more workers to match this $600 Billion increase, so they begin touring colleges, universities and high schools looking to recruit all sorts of people into NASA projects. STEM education funding goes high up, and motivation in global schools to go into degrees related to this field go up to levels higher than the 1960s. Kids want to be like one of those astronauts going to Phobos.
A lot more people are hired and a lot of people are motivated and newly optimistic of humanity itself. Space Agencies around the world want to work with NASA even more, if only to get a small slice of the tremendous budget. NASA, if its feeling nice enough, decides to bump up ESA's $5 billion annual budge to $10 billion, increases partnerships internationally, and maybe even wins the Russian Space Agency over (that or there is a huge brain drain in Russia that goes on for years).
Some of the money trickles back into other agencies. The Department of Energy, Department of Education, Department of Health (or whatever that is), Department of Commerce, obviously NASA's aeronautics division which is focused on airplanes, EPA, Congress (increased space lobbying that gets convincing bipartisan support), and obviously the Military.
The 2017 fiscal budget is changed back, but NASA now holds up to 2.5-3% of the federal budget, and the Military holds just a little bit less. The Military is able to resume its projects, reopen branches, and able to get on as normal, doing some more DARPA collaboration with NASA. NASA's $600 Billion slowly decreases, eventually coming back down to normal levels after 20-25 years. If they were smart and decided to lead several private spaceflight projects, they have the private industry to fall back on to continue to do amazing things. If not, then they are still in a bit better of a position than now, but there is still that same disappointment that their glory days are behind them and they must focus on scaling back.
Many more advancements will be made in the years that follow, and the world will come to recognize the $600 Billion anamoly as one of the monumental events in US History.
