What could a doubling of the NASA budget do? All kinds of things. It would be a good down-payment on Mars (even more so if we keep that spending level up). We are seeing a movement towards private space exploration, which is a very important trend. We could establish lots of prize funds with an extra $15 billion or so.
The last time NASA's budget was doubled from its steady state level we did great things. Here is the budget of NASA since 1958 in constant 2007 dollars:
When we made it to the moon in the late sixties, we were essentially doing it from scratch. We have much more groundwork laid today. If we could make this switch from subsidizing activity we have done for thousands of years to subsidizing frontier-expanding activity and maintain that higher level of spending in a deficit neutral way, we wouldn't have an Apollo program spike. We would have year after year of effort approaching what this endeavor deserves.
Eliminate all agricultural subsidies and double NASA's budget - spread the word, people.
Nobody has expressed the logic of overcoming externalities more eloquently than Armstrong: "one small step for man, one giant leap for mankind".