Can I live off post-docs? I'm currently a theoretical/mathematical physics PhD student in Brazil. I still have little more than a year until my PhD finishes, and as far as tradition goes, it's soon time to look for a post-doc position somewhere.

Brazil has recently suffered a coup d'état, with its legitimate president being replaced by a puppet. One of his most significant measures was to reduce science funding to almost zero, guaranteeing the best Brazilian researches would leave the country. In the end, staying here is unfeasible for a scientist, which means I'll be applying to Europe. I am also an European citizen, which means I can stay there for as long as I want.

Now, I'm also a very active climber. Climbing pretty much shapes my life in almost every sense. I love travelling and I'm used to spending almost nothing so that I can climb as much as possible. I was talking to a friend today about a plan of going for as many post-doc positions in Europe as I can, preferably near places I can climb, and keep my routine of climbing &amp; doing physics - that is, I want to avoid dealing with the mess a fixed position comes with: classes, bureaucracy, etc. In the end he said that this plan would not work after the third or fourth post-doc: I would become too expensive to hire due to accumulated degrees, and people would only find hiring me advantageous if I were extremely good. The point is that he himself went for 5 post-docs, and I know people who went even further. They quit because were offered a fixed position and took it... But can I live like that "forever"?
It will be very difficult to do so. Several European countries have rules about how long someone can be employed in "government" positions without a permanent contract. For instance, in Germany, if you do not have a unbefristete (permanent) contract after a given time, you are required to leave university employment. Other countries almost certainly have similar rules, as mentioned in the comments.

Now, there are positions above the postdoc level where one can remain indefinitely. For instance, in Germany, there are positions like the Akademischer Rat or Oberingenieur or außerplanmäßige Professor, who act in a secondary role supporting the head of an institute. Such positions involve substantially more administrative work than a postdoc normally encounters, but are also not subject to the time limitations. In other countries, such as the US, you can have "soft-money" positions that go by titles such as "research assistant professor" that are not formal tenure-track positions, and are dependent on the supervising faculty and the RAP raising sufficient funds to "pay for itself." I know multiple people who have had careers in such positions.