How to maximize shelf-life of freeze-dried food beyond 25 years? How far can this be exceeded?

The question is rather difficult to answer since it is based on false assumptions. I will try addressing those.

What process happens after 25-30 years which limits its shelf life?

There is no reason to assume that there is such a process. Imagine that I tell you "tomorrow at 2 PM, I will be at home". This contains no information about whether I will be at home tomorrow until 2:15, or until 6, or any other chosen time after 2 PM. Similarly, a "best by" date of 25 years makes absolutely no claims for the food quality at any time point after 25 years have passed. The existing foods may still taste good enough after 50 or 100 years.

if one wanted to increase the shelf-life to say 50 years, maybe even 100 years. What food break down processes will be encountered as the limiting factor

The assumption here is more subtle: it equates "food break down processes" with "the limiting factor". But it is also quite certainly false. It is not clear if there are any food break down processes that prevent longer shelf life. But if there are some, it is highly unlikely that they are also limiting factors. The hardest limiting factors you are likely to face are:

1. Developing a testing process that can prove that the food keeps its quality for 100 years with reasonable certainty, without actually waiting 100 years. This could be possible, but it will be quite hard. It will certainly not be as easy as creating such a process for something with a much better describable behaviour, such as electronics.
2. Convincing the FDA or other relevant regulators that your test results are correct
3. Establishing yourself in an already filled small niche market, despite having higher production costs than the incumbents (due to points 1 and 2).

I don't have answers to the other subquestions, but as I said, they are most likely based on a false assumption and thus unanswerable. If it turns out that they do have an answer in principle, the points above still render them pretty much irrelevant for practical purposes (which in itself is difficult enough).

I agree with rumtscho's answer, believing the logic behind it.

There is however one factor that could possibly have an effect, hinted at in the question: gas permeability of the package. They assume it's stored clean and dry, so we'll neglect water vapour. Nitrogen and CO2 are inert for our purposes, which leaves oxygen. Oxygen causes flavour molecules to degrade; it doesn't magically make food dangerous, so we're talking about quality not safety here.

The question says there's an oxygen scavenger in there. A common type is mainly iron, which rusts, capturing oxygen, and using up the iron. There are other types, but all have a finite ability to absorb O2. A large part of the purpose may well be to remove residual oxygen that gets in during packing - oxygen will be absorbed into even vacuum-dried food, and will work its way out, though that won't take decades. The packaging, despite being sealed, is slightly porous to gas molecules - not as porous as a rubber balloon which goes flat in a few days, but some will leak (this is why better helium balloons are foiled - He atoms are smaller than O2 molecules and get through more easily).

It is possible that they've run the numbers on the porosity of the packaging materials compared with the capacity of the oxygen scavenger, and decided that in 25 years it won't saturate.

As with other degradation processes, this doesn't mean that in 26 years it will spoil, just that if you and the manufacturer are still around in 25 years and it tastes worse than it did to start with, tough luck.