Until the end of the 19th century, nearly all science was done by amateurs. The first modern research university was Johns Hopkins, founded in 1876. Up until that time, universities were vehicles for transmitting established knowledge – divinely revealed wisdom at first, the humanistic classics later. They were not seen as engines for creating and discovering knowledge until well into the 20th century.
Even then, the primary responsibility of professors was teaching. Research was a side hustle, funded out of the professor’s salary or the occasional philanthropic bequest. Most full-time professional researchers worked at independent institutes like the Rockefeller Institute (now University) or the Kaiser Wilhelm (now Max Planck) Institutes in Germany. Or at the new corporate behemoths that began forming in the early 20th century, like GE or Hoechst or Bayer.
The modern research university, which has come to dominate scientific research, was largely the creation of one man – Vannevar Bush – who, after leading the Manhattan Project during WWII, insisted that US security and dominance required sustained government funding of basic science. The resulting National Science Foundation began supporting research by university professors through competitive grants, a model which was adopted also by the National Institutes for Health.
Between the space race and, later, the War on Cancer, these grants became a significant source of income to US universities, who routinely charge a 40% overhead on all grants won by professors. And the professors salaries are also paid in full or part by these grants. Star professors thus became cash cows for universities, and were relieved of much or even all of their teaching duties.
But I digress.
Astronomy is probably the field in which amateurs most commonly make significant contributions. Most comets have been discovered by amateurs, although this is changing as Big Data sky surveys are implemented. The thing about the sky is that it is accessible to everyone everywhere.
From comet hunter Colored dots are comets found by amateurs
In biology, the chemist turned attorney Gunter Wachterhauser proposed an intriguing hypothesis for the origin of life based on sulfur-iron redox chemistry. 
Israel Ramirez has written a nice explainer of this scheme.
The next significant contributor who comes to mind is Peter Mitchell, who developed the chemisosmotic theory of biological energy generation. But even he is borderline for being considered an amateur.
Energy conversion by the inner mitochondrial membrane and chemiosmotic coupling between the chemical energy of redox reactions in the respiratory chain and the oxidative phosphorylation catalysed by the ATP synthase. By Chiswick Chap – Own work, CC BY-SA 4.0
Margie Profet developed several significant hypotheses about the evolution of human female reproductive biology – that menstruation is a defense against pathogens introduced by frequent sexual activity, and that morning sickness is a defense against developmental defects in embryos caused by food toxins. Although she had some university affiliations, I don’t believe she was drawing a salary when she developed these ideas.
The professionalization of science greatly reduced the opportunity for amateurs to contribute. The fruit of the tree of science has been picked pretty clean on the lower branches, and most of the middle and higher ones as well. It’s usually a long arduous climb to find something new and untouched, and few amateurs have the time and resources required.
The best bets for amateur scientists are in fields where direct observation leads to discovery. There are still lots of beetles out there waiting to be found and described. There are even more bacteria and viruses, and as the cost of sequencing drops, I suspect microbiology might become a productive field for amateur investigation.
As of late, we look at the “Dunning Krueger curve” as a negative thing (specifically referring to over confident, arrogant know-nothing’s in government). But maybe there is far more value in the confidence vs experience curve when thinking of scientific achievement. Novices take risks that more experienced scientists may not. They (can) propagate abstract thought where experts may find barriers against. They force competition (in a healthy way) in areas we may not have thought about……catalyzing discovery in new areas by more well-resourced institutions.
Progressive Achievement is iterative and collaborative. We need contributions from both ends of the “DK” curve.
Thanks, Drew
Yeah – big advances, by definition, are doing things that experts consider highly unlikely if not impossible. Of course, the experts are almost always right. But that “almost” is a critical qualifier.