Can good, or cool, science research and inventing (R&D) be done on a budget with out the goverments money or control?

You need materials, you need equipment, you need a laboratory, you likely need research associates to help you on site, if you are working with animals you need to house them, feed them and acquire them in the first place, all of which costs money, and that is after you pay for living expenses.
Without any outside funding that is large enough to pay for both living expenses and research, you can only do research in your spare time after your dayjob, slowing you down massively, the low budget limits you to low tier materials and equipment, and not having helpers on site and only on the web with time delayed communication slows you down even more.
At the very least, it makes you completely uncompetitive against publicly or corporate funded researchers, because they will be faster than you, and because they have access to specialised and expensive equipment, they can venture into fields that you never will.

Government funding allows you the freedom to take the risk. A lot of time and resources need to be invested on projects that only have a small chance of success. Government funding often covers “high risk, high reward” research.

It also gives small, agile companies access to labs, facilities, and equipment typically beyond their reach.

The money also comes from non-military sources, like Department of Agriculture. Research needs a lot of documentation and that costs money.

Kickstarter.com It’s pretty well know so you might have heard of it.

Don’t forget R&D done by NASA and NACA. That 0.5% of the National Budget got us some cool stuff.

Iam sure computer hackers do it all the time, for free.