Can I live off post-docs?

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.

Can I live off post-docs?

I disagree with the other answers, based on my own experience my answer is yes. However it's unusual and it has some negative sides.

I have been a postdoc in France and Ireland for 12 years, working in three different academic institutions.

Let's not kid ourselves: the other answers are correct in saying that this is a bad career path (assuming you want a real academic career), and the uncertainty about still having a job next year is not ideal in everyday life. Additionally, usually the salary scale for a postdoc will not progress a lot (I assume that this depends on the country).

However there is a strong need for skilled researchers on a temporary basis in many places. It is true that many countries in Europe have limits on the number of years you can stay on a temporary contract, but (1) not all of them and (2) some institutions simply turn a blind eye on the legislation. Since you seem to be ok with regularly moving from one place to another, it looks like this wouldn't be an obstacle in your case.

Research wise, it's hard to have your own long term research plan and make it progress this way. On the other hand, you have less administrative and teaching duties, so more time to do actual research.

And in case you are bored with this one day, normally it wouldn't be too hard to have job offers from the private sector.

No, please don't even think about it.

Some fields do need and have longer median postdoctoral times and stints (e.g. 2 postdocs are common in the biological sciences), but it is essentially career destroying if you're shuffling around from one postdoc to the other.

There are many reasons.

(I) Depreciation of one's academic worth with time when one lingers on in a postdoctoral role is very real.

(II) Any cool work that you do as a postdoc, or great papers that you write (with the rare exception of pure math, where authorship works differently) is invariably credited to the PI of the research group you're postdocing with.

(III) If you want a pure "scientist" job without the rest of the hassle that being a faculty member entails, how about applying for permanent positions at research institutions? That would be the equivalent of places like NIST, GFDL or Brookhaven or other National Labs in the US. It also pays way better than a postdoc.

(IV) Lastly, anecdotes about "career" postdocs are none too encouraging. Here is one: A brilliant researcher "X" worked with a famous PI, Prof. "Y" as a postdoc. "X" was technically amazing, and could set up certain difficult experiments and measurements like no one else.

It was very useful for "Y" to have "X" around, so he did. "X" had a great time, stayed on with "Y" as a postdoc, and got co-authorship on papers written in top journals. Unfortunately, "X" got so comfortable in her position that she did not realize the clock was ticking away.

Until one fine day, when Prof. "Y" decided to switch research areas completely, and told a rather middle-aged "X" to find a new position elsewhere.