Well...I think what we actually mean is if the budget were doubled from its current $14 billion to $28 billion, and that this funding level remained constant in REAL CURRENT DOLLARS, then what would we be able to do in space?
This is a tough one to speculate upon. I think that you would see a dramatic increase in the number of happy astronomers with long-term job stability! You would see many more very ambitious engineering and scientific projects to detect life and planets orbiting other stars. There would be a full fledged lunar research outpost with some very impressive telescopes in operation. The list is very long as any science fiction reader might tell you. The BIG problems are still in two very specific areas.
Do we as a nation really have the will to undertake ambitious long term programs? During the last 20 years we have spent endless time 'debating' why we need a Space Station, why we 'really' need to go to the Moon and Mars etc. While the debating goes on, we loose precious opportunities to carry out these projects at lower cost than what we now have to pay. I think these kind of debates will escalate into full-fledged stalemates of research opportunity. When the USSR was still our bogyman, the debate was about political and security advantages. But today, it is infinitely harder to convince anyone to do anything that costs money. Every ambitious undertaking is instantly mired in a debate over 'why' we need to do this. To me, it sounds very much like students asking why they have to do this particular homework assignment. No matter how much you show charts and evidence that a dollar invested in space returns 5 - 10 dollars on the ground, no one seems to want to be compelled even by financial arguments.
The second problem is that our basic understanding of how to carry out large projects is still rudimentary. We do not know how to put someone in space for more than 200 days without serious medical impacts. We do not know how to build a closed ecosphere for long term habitation. Our technologies for putting a kilogram of material in orbit still costs thousands of dollars a pound. Even if we had more money, it is not obvious how to accelerate the learning process, but it would sure help to have more resources and people involved. In a Catch-22 situation, we will not get the experience we need until we invest in the basic research we need to get us over the technological learning curve. But we cannot seem to convince anyone that without reliable support and investment, we will never be able to solve the big problems that still daunt us. The returns on these investments are in the future, and this is where the root of the problem lies.