Academic alternatives to a Ph.D "You became a professor at Cornell without ever having received a
  Ph.D. You seem almost proud of that fact.
  
  Oh, yes. I’m very proud of not having a Ph.D. I think the Ph.D. system
  is an abomination. It was invented as a system for educating German
  professors in the 19th century, and it works well under those
  conditions. It’s good for a very small number of people who are going
  to spend their lives being professors. But it has become now a kind of
  union card that you have to have in order to have a job, whether it’s
  being a professor or other things, and it’s quite inappropriate for
  that. It forces people to waste years and years of their lives sort of
  pretending to do research for which they’re not at all well-suited. In
  the end, they have this piece of paper which says they’re qualified,
  but it really doesn’t mean anything. The Ph.D. takes far too long
  and discourages women from becoming scientists, which I consider a
  great tragedy. So I have opposed it all my life without any success
  at all.
  
  How is it that you were able to escape that requirement?
  
  I was lucky because I got educated in World War II and everything was
  screwed up so that I could get through without a Ph.D. and finish up
  as a professor. Now that’s quite impossible. So, I’m very proud that I
  don’t have a Ph.D. and I raised six children and none of them has a
  Ph.D., so that’s my contribution."


(From [1])

Which are the Academic alternatives to a Ph.D.? (Actively participate in seminars, write papers,) being supervised without earning a degree? How you become deeply involved in the community of your research interests without a Ph.D.? How you become known in your field? How do you write to a professor for supervision without earning a degree?

[1]: A 'rebel' without a Ph.D. A conversation with the mathematical physicist Freeman Dyson on quantum electrodynamics, climate change, and his latest pet project.
Freeman Dyson had two advantages on his side: (i) He was brilliant. (ii) He was educated at a time when, as he put it, everything was "screwed up".

(ii) is no longer the case for any of us, or at least not in any of the countries where I take it that you want to be a professor. (i) may be true for you, but statistically speaking is not true for all but a very very small number of people; consequently, (i) also does not provide a route to becoming a professor, and certainly is not something anyone should bank upon.

In other words: Forget about becoming a professor without a PhD. While it may not be completely impossible, it is so unlikely that nobody could reasonably advise anyone to take that route. In actual practice, almost all professor job ads contain a phrase similar to "PhD in X or a related field is required", and there truly have to be exceptional reasons for a hiring committee to ignore this.

[I will add that all of the above specifically only applies to the sciences and mathematics; professors are frequently hired out of industry in engineering, education, and business departments, and I would not be surprised if other criteria apply in such cases.]