The title isn't accurate, so I can't assume it is.
Fiat currency works precisely because the planet is finite. What you're thinking of is a currency that's tied to a finite resource, like a gold standard. Fiat currency works precisely because monetary supply is able to be increased to keep pace with both the population and economic growth. Likewise, fiat currency can be removed from the economy when the economy shrinks (although this hasn't happened before).
Gold isn't finite, its supply increases every year and is the main reason for the stability in its prices. Obviously this can't continue forever, hence the finite resources on the planet
Two things you're assuming is fiat can be evenly distributed with population growth and at some point the issuing body will reduce excess supply, neither of these things are common practices and rarely happen
Gold is the definition of finite. There is only so much gold in the world--by which I mean the 3rd planet revolving around the sun in the solar system that we both reside in--and the only way to make more gold actually makes radioactive gold. THe fact that we haven't discovered every last atom of it doesn't make it any less finite.
Different infinities of a particular finite resource do not exist though. Yes, we discover more every year, but that does not make it an infinite resource. If a country was dumb enough to link their monetary supply to a resource, and then their economic growth outstripped their domestic production of that resource, they would experience deflation, id est, the money you have now would be worth more tomorrow than it is today. Periods of deflation, while rare, always have a catastrophic effect on economies, because no one wants to spend money when the value of that money is increasing. People tend to hoard their money when that happens, which rapidly leads to recession and then depression.
Your core premise is, again, flawed. You want a good way of storing 'value'--which is a nebulous concept to start with--and then condition that premise on not using a fiat currency. Fiat currency is the best way of storing that value, and using that value at some point in the future. You could invest in stocks, bonds, money market accounts, or real estate, but that's all still based on a fiat currency of some kine.
Value is vague on purpose, since it means that something retains its "essence" tomorrow as good or better than today. Otherwise humans would not even be able to feed themselves since the only reason I can have food in the winter is because I have something to exchange for food from places that have figured out to grow or store food. It also would be a terrible place if we worked very hard today and our returns depreciated rapidly, because there will be times when we can't work
Sure, stocks and bonds have historically held some value, but is near impossible to out perform base inflation and has gotten worse the last 20+ years. Physical land and tangible long term goods have held up, but difficult to use for trade, hence currency
I'm pro-fiat because it works. Crypto is okay, but it's too volatile, and has too many unknowns, to be a good investment vehicle. Crypto is only useful if you're actively buying things. Real estate is illiquid. Gold can't be used readily without being converted to a fiat currency; you aren't going to be able to convince any grocery store to accept an ounce of gold in place of currency. Treasury bills and bonds nearly always outperform base inflation, even if it's not by very much. If I'm reading things correctly, the current rate on short-term US treasury bills is about 2% over inflation. Longer term bonds tend to have higher yields. Both are lower than you could expect to get with a money market or index fund. And an index fund is going to have lower yield than a growth fund.
Almost everything out-performs base inflation rates over the long term, because if it didn't, no one would invest.