In astronomy, would it be possible to continue doing postdocs for up to 15 years to increase your chance of getting a permanent position?
This question to some extent revolves around a purely semantic issue, namely what is a postdoc? Many areas in academia have a small number of "soft money" research positions that a researcher can hold, sometimes for many years. When a researcher has an independent source of funds and doesn't need to be paid, I imagine that can only improve his/her ability to obtain such positions. The questions that need to be discussed in connection to your question are therefore:

(a) Do such positions qualify as "postdocs"?

(b) Is there a chance for a researcher holding such a position for up to 15 years to eventually transfer to a tenure track position?

I'm not in astronomy, but from a general STEM perspective I would argue that the answer to (a) is no, and the answer to (b) is yes.

Specifically, with regards to (a), the point is that "postdoc" is not a technical term (at least not one whose definition is universally agreed upon), but an informal label we apply to an interim position a researcher holds for a few years after finishing their PhD and before finding a permanent/tenure-track position. Talking about someone being a postdoc for 15 years is an oxymoron, as it stretches the term way past its conventionally understood meaning. Linguistically it would be like talking about a "50 year old child."

With regards to (b), I think it is indeed very possible for a researcher to transition to a faculty position from any other type of position, provided they have been doing high quality work. An extreme case (but not necessarily the only case) would be a researcher who makes a single very important scientific discovery that propels them to scientific stardom. I would imagine that it is absolutely possible for this to happen to a non-faculty researcher in astronomy. Whether it's likely to happen is a completely different story of course, but given that the premise of OP's question is that landing a faculty position in astronomy is already highly unlikely even in the normal way, I don't see why this approach is any less likely to work than the more conventional one.

To summarize, my (non-astronomy-specific, as I said) opinion is that although a soft money researcher may suffer from a certain minor stigma when it comes to applying for faculty positions, this by itself will almost certainly not be a deal-breaker, and when it comes to being evaluated for such positions, to a good degree of approximation everything would depend on the work they have published.