What missions could NASA persue if we increased their budget 10 fold.?

Even with a tenfold increase there will STILL not be a manned mission to Mars.

NASA will not willfully kill astronauts and that is exactly what a manned Mars mission would do. Radiation sickness, bone density loss, tooth loss, gut problems, - humans can not survive that long in micro gravity.

To get pseudo gravity you need a much bigger and spinning spacecraft. At that size the metalwork is scattering too much radiation so now you have to add in a jacket of ice about a foot thick over the whole structure. The mass becomes phenomenal and well beyond the lifting capacity of NASA with fifty times the funding.

A Mars mission would be completely possible. You don't need a huge rotating ring to solve the problem of zero gravity, you only need two parts of a spacecraft separated by a tether and spun during the 6-month trip there and another one back a year later. In any case, humans have spent more than 6 months in micro gravity without any permanent damage.

The National Aeronautics and Space Administration is a manager. Corporations, universities, global scientific organizations, etc. propose projects and concepts to be studies. NASA is not a mission generator but a mission producer. Oh, there are phone numbers listed that you can call and ask.

more satellites. well have a dome on the moon to do research there. we really need a new telescope.

They could make a really great film.